Properties

Degree 38
Conductor $ 2^{38} \cdot 19^{19} \cdot 53^{19} $
Sign $-1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 19

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 2·5-s − 11·7-s − 11·9-s − 11·11-s + 13-s + 8·15-s − 17-s − 19·19-s + 44·21-s − 16·23-s − 35·25-s + 68·27-s − 9·31-s + 44·33-s + 22·35-s − 25·37-s − 4·39-s + 41-s − 41·43-s + 22·45-s − 29·47-s + 49-s + 4·51-s + 19·53-s + 22·55-s + 76·57-s + ⋯
L(s)  = 1  − 2.30·3-s − 0.894·5-s − 4.15·7-s − 3.66·9-s − 3.31·11-s + 0.277·13-s + 2.06·15-s − 0.242·17-s − 4.35·19-s + 9.60·21-s − 3.33·23-s − 7·25-s + 13.0·27-s − 1.61·31-s + 7.65·33-s + 3.71·35-s − 4.10·37-s − 0.640·39-s + 0.156·41-s − 6.25·43-s + 3.27·45-s − 4.23·47-s + 1/7·49-s + 0.560·51-s + 2.60·53-s + 2.96·55-s + 10.0·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{38} \cdot 19^{19} \cdot 53^{19}\right)^{s/2} \, \Gamma_{\C}(s)^{19} \, L(s)\cr =\mathstrut & -\,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{38} \cdot 19^{19} \cdot 53^{19}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{19} \, L(s)\cr =\mathstrut & -\,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(38\)
\( N \)  =  \(2^{38} \cdot 19^{19} \cdot 53^{19}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{4028} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  19
Selberg data  =  $(38,\ 2^{38} \cdot 19^{19} \cdot 53^{19} ,\ ( \ : [1/2]^{19} ),\ -1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;53\}$, \(F_p\) is a polynomial of degree 38. If $p \in \{2,\;19,\;53\}$, then $F_p$ is a polynomial of degree at most 37.
$p$$F_p$
bad2 \( 1 \)
19 \( ( 1 + T )^{19} \)
53 \( ( 1 - T )^{19} \)
good3 \( 1 + 4 T + p^{3} T^{2} + 28 p T^{3} + 341 T^{4} + 886 T^{5} + 2785 T^{6} + 6377 T^{7} + 634 p^{3} T^{8} + 35819 T^{9} + 86396 T^{10} + 169078 T^{11} + 125701 p T^{12} + 233291 p T^{13} + 489352 p T^{14} + 2603456 T^{15} + 576676 p^{2} T^{16} + 8831093 T^{17} + 16849145 T^{18} + 27567853 T^{19} + 16849145 p T^{20} + 8831093 p^{2} T^{21} + 576676 p^{5} T^{22} + 2603456 p^{4} T^{23} + 489352 p^{6} T^{24} + 233291 p^{7} T^{25} + 125701 p^{8} T^{26} + 169078 p^{8} T^{27} + 86396 p^{9} T^{28} + 35819 p^{10} T^{29} + 634 p^{14} T^{30} + 6377 p^{12} T^{31} + 2785 p^{13} T^{32} + 886 p^{14} T^{33} + 341 p^{15} T^{34} + 28 p^{17} T^{35} + p^{20} T^{36} + 4 p^{18} T^{37} + p^{19} T^{38} \)
5 \( 1 + 2 T + 39 T^{2} + 82 T^{3} + 749 T^{4} + 1641 T^{5} + 1908 p T^{6} + 21342 T^{7} + 91512 T^{8} + 202596 T^{9} + 708864 T^{10} + 1499102 T^{11} + 924909 p T^{12} + 9095878 T^{13} + 26247629 T^{14} + 47825634 T^{15} + 134993734 T^{16} + 9356587 p^{2} T^{17} + 666823791 T^{18} + 1147399136 T^{19} + 666823791 p T^{20} + 9356587 p^{4} T^{21} + 134993734 p^{3} T^{22} + 47825634 p^{4} T^{23} + 26247629 p^{5} T^{24} + 9095878 p^{6} T^{25} + 924909 p^{8} T^{26} + 1499102 p^{8} T^{27} + 708864 p^{9} T^{28} + 202596 p^{10} T^{29} + 91512 p^{11} T^{30} + 21342 p^{12} T^{31} + 1908 p^{14} T^{32} + 1641 p^{14} T^{33} + 749 p^{15} T^{34} + 82 p^{16} T^{35} + 39 p^{17} T^{36} + 2 p^{18} T^{37} + p^{19} T^{38} \)
7 \( 1 + 11 T + 120 T^{2} + 895 T^{3} + 6137 T^{4} + 35639 T^{5} + 191013 T^{6} + 925604 T^{7} + 4190306 T^{8} + 17643355 T^{9} + 70149327 T^{10} + 263386198 T^{11} + 941475306 T^{12} + 3207913105 T^{13} + 1495440960 p T^{14} + 32760289701 T^{15} + 98594518373 T^{16} + 285642829543 T^{17} + 797795552662 T^{18} + 2148879348566 T^{19} + 797795552662 p T^{20} + 285642829543 p^{2} T^{21} + 98594518373 p^{3} T^{22} + 32760289701 p^{4} T^{23} + 1495440960 p^{6} T^{24} + 3207913105 p^{6} T^{25} + 941475306 p^{7} T^{26} + 263386198 p^{8} T^{27} + 70149327 p^{9} T^{28} + 17643355 p^{10} T^{29} + 4190306 p^{11} T^{30} + 925604 p^{12} T^{31} + 191013 p^{13} T^{32} + 35639 p^{14} T^{33} + 6137 p^{15} T^{34} + 895 p^{16} T^{35} + 120 p^{17} T^{36} + 11 p^{18} T^{37} + p^{19} T^{38} \)
11 \( 1 + p T + 170 T^{2} + 1466 T^{3} + 13698 T^{4} + 97805 T^{5} + 703713 T^{6} + 4316820 T^{7} + 26040453 T^{8} + 140683756 T^{9} + 740890277 T^{10} + 3584559765 T^{11} + 16853652949 T^{12} + 6713689690 p T^{13} + 28561797950 p T^{14} + 1256137321885 T^{15} + 4875103100411 T^{16} + 17865248014363 T^{17} + 63553609216286 T^{18} + 213932167946862 T^{19} + 63553609216286 p T^{20} + 17865248014363 p^{2} T^{21} + 4875103100411 p^{3} T^{22} + 1256137321885 p^{4} T^{23} + 28561797950 p^{6} T^{24} + 6713689690 p^{7} T^{25} + 16853652949 p^{7} T^{26} + 3584559765 p^{8} T^{27} + 740890277 p^{9} T^{28} + 140683756 p^{10} T^{29} + 26040453 p^{11} T^{30} + 4316820 p^{12} T^{31} + 703713 p^{13} T^{32} + 97805 p^{14} T^{33} + 13698 p^{15} T^{34} + 1466 p^{16} T^{35} + 170 p^{17} T^{36} + p^{19} T^{37} + p^{19} T^{38} \)
13 \( 1 - T + 127 T^{2} - 68 T^{3} + 626 p T^{4} - 1707 T^{5} + 353317 T^{6} + 1062 T^{7} + 11684909 T^{8} + 1560084 T^{9} + 312600446 T^{10} + 65083687 T^{11} + 7010909859 T^{12} + 1722230180 T^{13} + 134899780238 T^{14} + 34925506310 T^{15} + 2260526879991 T^{16} + 44714588876 p T^{17} + 33291077083366 T^{18} + 8182233606402 T^{19} + 33291077083366 p T^{20} + 44714588876 p^{3} T^{21} + 2260526879991 p^{3} T^{22} + 34925506310 p^{4} T^{23} + 134899780238 p^{5} T^{24} + 1722230180 p^{6} T^{25} + 7010909859 p^{7} T^{26} + 65083687 p^{8} T^{27} + 312600446 p^{9} T^{28} + 1560084 p^{10} T^{29} + 11684909 p^{11} T^{30} + 1062 p^{12} T^{31} + 353317 p^{13} T^{32} - 1707 p^{14} T^{33} + 626 p^{16} T^{34} - 68 p^{16} T^{35} + 127 p^{17} T^{36} - p^{18} T^{37} + p^{19} T^{38} \)
17 \( 1 + T + 150 T^{2} + 175 T^{3} + 11058 T^{4} + 13274 T^{5} + 540367 T^{6} + 599326 T^{7} + 19853003 T^{8} + 18456246 T^{9} + 588688177 T^{10} + 417341732 T^{11} + 14782761103 T^{12} + 7212484674 T^{13} + 326065477437 T^{14} + 97948236591 T^{15} + 6493166151252 T^{16} + 1135467731678 T^{17} + 118909992017306 T^{18} + 15315368153951 T^{19} + 118909992017306 p T^{20} + 1135467731678 p^{2} T^{21} + 6493166151252 p^{3} T^{22} + 97948236591 p^{4} T^{23} + 326065477437 p^{5} T^{24} + 7212484674 p^{6} T^{25} + 14782761103 p^{7} T^{26} + 417341732 p^{8} T^{27} + 588688177 p^{9} T^{28} + 18456246 p^{10} T^{29} + 19853003 p^{11} T^{30} + 599326 p^{12} T^{31} + 540367 p^{13} T^{32} + 13274 p^{14} T^{33} + 11058 p^{15} T^{34} + 175 p^{16} T^{35} + 150 p^{17} T^{36} + p^{18} T^{37} + p^{19} T^{38} \)
23 \( 1 + 16 T + 319 T^{2} + 3618 T^{3} + 43748 T^{4} + 394447 T^{5} + 3653016 T^{6} + 27822565 T^{7} + 215894329 T^{8} + 1446039217 T^{9} + 9889502217 T^{10} + 60060497332 T^{11} + 374474498134 T^{12} + 2110058255001 T^{13} + 12257266225486 T^{14} + 2824677544609 p T^{15} + 355248137995664 T^{16} + 1779048859027991 T^{17} + 9176892138573302 T^{18} + 43385874507698012 T^{19} + 9176892138573302 p T^{20} + 1779048859027991 p^{2} T^{21} + 355248137995664 p^{3} T^{22} + 2824677544609 p^{5} T^{23} + 12257266225486 p^{5} T^{24} + 2110058255001 p^{6} T^{25} + 374474498134 p^{7} T^{26} + 60060497332 p^{8} T^{27} + 9889502217 p^{9} T^{28} + 1446039217 p^{10} T^{29} + 215894329 p^{11} T^{30} + 27822565 p^{12} T^{31} + 3653016 p^{13} T^{32} + 394447 p^{14} T^{33} + 43748 p^{15} T^{34} + 3618 p^{16} T^{35} + 319 p^{17} T^{36} + 16 p^{18} T^{37} + p^{19} T^{38} \)
29 \( 1 + 263 T^{2} + 227 T^{3} + 32667 T^{4} + 57816 T^{5} + 2602535 T^{6} + 6904872 T^{7} + 153336076 T^{8} + 520219805 T^{9} + 7310658090 T^{10} + 28193282237 T^{11} + 299486963727 T^{12} + 1195035541881 T^{13} + 10947633789555 T^{14} + 42383215737484 T^{15} + 365629929365909 T^{16} + 1338432920697978 T^{17} + 11331129770587833 T^{18} + 39644618388421176 T^{19} + 11331129770587833 p T^{20} + 1338432920697978 p^{2} T^{21} + 365629929365909 p^{3} T^{22} + 42383215737484 p^{4} T^{23} + 10947633789555 p^{5} T^{24} + 1195035541881 p^{6} T^{25} + 299486963727 p^{7} T^{26} + 28193282237 p^{8} T^{27} + 7310658090 p^{9} T^{28} + 520219805 p^{10} T^{29} + 153336076 p^{11} T^{30} + 6904872 p^{12} T^{31} + 2602535 p^{13} T^{32} + 57816 p^{14} T^{33} + 32667 p^{15} T^{34} + 227 p^{16} T^{35} + 263 p^{17} T^{36} + p^{19} T^{38} \)
31 \( 1 + 9 T + 268 T^{2} + 2260 T^{3} + 36981 T^{4} + 291295 T^{5} + 3491376 T^{6} + 25906909 T^{7} + 254051465 T^{8} + 1790063164 T^{9} + 15199900498 T^{10} + 102054137753 T^{11} + 775688214172 T^{12} + 4962816000933 T^{13} + 34475251192351 T^{14} + 209822120776297 T^{15} + 1350769017614591 T^{16} + 7801487741761741 T^{17} + 46992591773508464 T^{18} + 256770621749748689 T^{19} + 46992591773508464 p T^{20} + 7801487741761741 p^{2} T^{21} + 1350769017614591 p^{3} T^{22} + 209822120776297 p^{4} T^{23} + 34475251192351 p^{5} T^{24} + 4962816000933 p^{6} T^{25} + 775688214172 p^{7} T^{26} + 102054137753 p^{8} T^{27} + 15199900498 p^{9} T^{28} + 1790063164 p^{10} T^{29} + 254051465 p^{11} T^{30} + 25906909 p^{12} T^{31} + 3491376 p^{13} T^{32} + 291295 p^{14} T^{33} + 36981 p^{15} T^{34} + 2260 p^{16} T^{35} + 268 p^{17} T^{36} + 9 p^{18} T^{37} + p^{19} T^{38} \)
37 \( 1 + 25 T + 751 T^{2} + 13385 T^{3} + 243505 T^{4} + 93159 p T^{5} + 47858705 T^{6} + 568669912 T^{7} + 6550582652 T^{8} + 67507394756 T^{9} + 672343075530 T^{10} + 6136000248927 T^{11} + 54107266908095 T^{12} + 443364156625774 T^{13} + 3512654394037347 T^{14} + 26079561106234904 T^{15} + 187358642326732771 T^{16} + 1267484363772225738 T^{17} + 8301592548220343267 T^{18} + 51315013186076839280 T^{19} + 8301592548220343267 p T^{20} + 1267484363772225738 p^{2} T^{21} + 187358642326732771 p^{3} T^{22} + 26079561106234904 p^{4} T^{23} + 3512654394037347 p^{5} T^{24} + 443364156625774 p^{6} T^{25} + 54107266908095 p^{7} T^{26} + 6136000248927 p^{8} T^{27} + 672343075530 p^{9} T^{28} + 67507394756 p^{10} T^{29} + 6550582652 p^{11} T^{30} + 568669912 p^{12} T^{31} + 47858705 p^{13} T^{32} + 93159 p^{15} T^{33} + 243505 p^{15} T^{34} + 13385 p^{16} T^{35} + 751 p^{17} T^{36} + 25 p^{18} T^{37} + p^{19} T^{38} \)
41 \( 1 - T + 550 T^{2} - 945 T^{3} + 146374 T^{4} - 349942 T^{5} + 25219659 T^{6} - 75244426 T^{7} + 3174543670 T^{8} - 10979362017 T^{9} + 311959036558 T^{10} - 1180422294000 T^{11} + 24917954256776 T^{12} - 98297563717089 T^{13} + 1658534291568757 T^{14} - 159726245990962 p T^{15} + 93342246610226873 T^{16} - 356435642476819079 T^{17} + 4476526560391372765 T^{18} - 16040172963957118964 T^{19} + 4476526560391372765 p T^{20} - 356435642476819079 p^{2} T^{21} + 93342246610226873 p^{3} T^{22} - 159726245990962 p^{5} T^{23} + 1658534291568757 p^{5} T^{24} - 98297563717089 p^{6} T^{25} + 24917954256776 p^{7} T^{26} - 1180422294000 p^{8} T^{27} + 311959036558 p^{9} T^{28} - 10979362017 p^{10} T^{29} + 3174543670 p^{11} T^{30} - 75244426 p^{12} T^{31} + 25219659 p^{13} T^{32} - 349942 p^{14} T^{33} + 146374 p^{15} T^{34} - 945 p^{16} T^{35} + 550 p^{17} T^{36} - p^{18} T^{37} + p^{19} T^{38} \)
43 \( 1 + 41 T + 1210 T^{2} + 26774 T^{3} + 501650 T^{4} + 8133183 T^{5} + 118151216 T^{6} + 1556708821 T^{7} + 18890009404 T^{8} + 212668330533 T^{9} + 2239932400711 T^{10} + 22175457407977 T^{11} + 207400542113396 T^{12} + 1838180140236470 T^{13} + 15488364055406778 T^{14} + 124314080562234228 T^{15} + 952381712403182927 T^{16} + 6972065687450028683 T^{17} + 48826921600417729150 T^{18} + \)\(32\!\cdots\!51\)\( T^{19} + 48826921600417729150 p T^{20} + 6972065687450028683 p^{2} T^{21} + 952381712403182927 p^{3} T^{22} + 124314080562234228 p^{4} T^{23} + 15488364055406778 p^{5} T^{24} + 1838180140236470 p^{6} T^{25} + 207400542113396 p^{7} T^{26} + 22175457407977 p^{8} T^{27} + 2239932400711 p^{9} T^{28} + 212668330533 p^{10} T^{29} + 18890009404 p^{11} T^{30} + 1556708821 p^{12} T^{31} + 118151216 p^{13} T^{32} + 8133183 p^{14} T^{33} + 501650 p^{15} T^{34} + 26774 p^{16} T^{35} + 1210 p^{17} T^{36} + 41 p^{18} T^{37} + p^{19} T^{38} \)
47 \( 1 + 29 T + 909 T^{2} + 18088 T^{3} + 351005 T^{4} + 5488261 T^{5} + 82311862 T^{6} + 1078604038 T^{7} + 13540700542 T^{8} + 3280825921 p T^{9} + 1685805736685 T^{10} + 17060978901367 T^{11} + 166174382371186 T^{12} + 1516221634325367 T^{13} + 13342302214277328 T^{14} + 110776535216503272 T^{15} + 888349952423380237 T^{16} + 6749274790948988982 T^{17} + 49571711840322407256 T^{18} + \)\(34\!\cdots\!10\)\( T^{19} + 49571711840322407256 p T^{20} + 6749274790948988982 p^{2} T^{21} + 888349952423380237 p^{3} T^{22} + 110776535216503272 p^{4} T^{23} + 13342302214277328 p^{5} T^{24} + 1516221634325367 p^{6} T^{25} + 166174382371186 p^{7} T^{26} + 17060978901367 p^{8} T^{27} + 1685805736685 p^{9} T^{28} + 3280825921 p^{11} T^{29} + 13540700542 p^{11} T^{30} + 1078604038 p^{12} T^{31} + 82311862 p^{13} T^{32} + 5488261 p^{14} T^{33} + 351005 p^{15} T^{34} + 18088 p^{16} T^{35} + 909 p^{17} T^{36} + 29 p^{18} T^{37} + p^{19} T^{38} \)
59 \( 1 + 42 T + 1359 T^{2} + 31896 T^{3} + 654747 T^{4} + 11517935 T^{5} + 185139300 T^{6} + 2692451456 T^{7} + 36557880014 T^{8} + 460650413958 T^{9} + 5487089427876 T^{10} + 61508874932730 T^{11} + 656823640582863 T^{12} + 6655930253502458 T^{13} + 64566281722336627 T^{14} + 597399001750638236 T^{15} + 5307312423594061832 T^{16} + 45105897961474589951 T^{17} + \)\(36\!\cdots\!47\)\( T^{18} + \)\(28\!\cdots\!68\)\( T^{19} + \)\(36\!\cdots\!47\)\( p T^{20} + 45105897961474589951 p^{2} T^{21} + 5307312423594061832 p^{3} T^{22} + 597399001750638236 p^{4} T^{23} + 64566281722336627 p^{5} T^{24} + 6655930253502458 p^{6} T^{25} + 656823640582863 p^{7} T^{26} + 61508874932730 p^{8} T^{27} + 5487089427876 p^{9} T^{28} + 460650413958 p^{10} T^{29} + 36557880014 p^{11} T^{30} + 2692451456 p^{12} T^{31} + 185139300 p^{13} T^{32} + 11517935 p^{14} T^{33} + 654747 p^{15} T^{34} + 31896 p^{16} T^{35} + 1359 p^{17} T^{36} + 42 p^{18} T^{37} + p^{19} T^{38} \)
61 \( 1 - T + 517 T^{2} + 241 T^{3} + 133255 T^{4} + 245250 T^{5} + 23188299 T^{6} + 70944406 T^{7} + 3102873365 T^{8} + 12517695983 T^{9} + 343125143088 T^{10} + 1609447575084 T^{11} + 32729710941875 T^{12} + 163931903376839 T^{13} + 2760204624527518 T^{14} + 13916645083482646 T^{15} + 208309095297321825 T^{16} + 1019676457461419921 T^{17} + 14128281306980348077 T^{18} + 65943742910952541230 T^{19} + 14128281306980348077 p T^{20} + 1019676457461419921 p^{2} T^{21} + 208309095297321825 p^{3} T^{22} + 13916645083482646 p^{4} T^{23} + 2760204624527518 p^{5} T^{24} + 163931903376839 p^{6} T^{25} + 32729710941875 p^{7} T^{26} + 1609447575084 p^{8} T^{27} + 343125143088 p^{9} T^{28} + 12517695983 p^{10} T^{29} + 3102873365 p^{11} T^{30} + 70944406 p^{12} T^{31} + 23188299 p^{13} T^{32} + 245250 p^{14} T^{33} + 133255 p^{15} T^{34} + 241 p^{16} T^{35} + 517 p^{17} T^{36} - p^{18} T^{37} + p^{19} T^{38} \)
67 \( 1 + 41 T + 1396 T^{2} + 32847 T^{3} + 682311 T^{4} + 11742555 T^{5} + 184509800 T^{6} + 2556216527 T^{7} + 33066576689 T^{8} + 389631495081 T^{9} + 4366480554048 T^{10} + 45585580889627 T^{11} + 460355414139920 T^{12} + 4406910850541114 T^{13} + 41392844964425483 T^{14} + 372892643057838574 T^{15} + 3324013680677058857 T^{16} + 28558548940799366476 T^{17} + \)\(24\!\cdots\!83\)\( T^{18} + \)\(20\!\cdots\!64\)\( T^{19} + \)\(24\!\cdots\!83\)\( p T^{20} + 28558548940799366476 p^{2} T^{21} + 3324013680677058857 p^{3} T^{22} + 372892643057838574 p^{4} T^{23} + 41392844964425483 p^{5} T^{24} + 4406910850541114 p^{6} T^{25} + 460355414139920 p^{7} T^{26} + 45585580889627 p^{8} T^{27} + 4366480554048 p^{9} T^{28} + 389631495081 p^{10} T^{29} + 33066576689 p^{11} T^{30} + 2556216527 p^{12} T^{31} + 184509800 p^{13} T^{32} + 11742555 p^{14} T^{33} + 682311 p^{15} T^{34} + 32847 p^{16} T^{35} + 1396 p^{17} T^{36} + 41 p^{18} T^{37} + p^{19} T^{38} \)
71 \( 1 + 902 T^{2} + 1085 T^{3} + 392410 T^{4} + 949536 T^{5} + 110022826 T^{6} + 400348787 T^{7} + 22430153668 T^{8} + 108208741832 T^{9} + 3558817249441 T^{10} + 21038930940961 T^{11} + 459400689288747 T^{12} + 3127977931159775 T^{13} + 49778932803940384 T^{14} + 368691409348339198 T^{15} + 4628758875972824499 T^{16} + 35206980919949657196 T^{17} + \)\(37\!\cdots\!96\)\( T^{18} + \)\(27\!\cdots\!60\)\( T^{19} + \)\(37\!\cdots\!96\)\( p T^{20} + 35206980919949657196 p^{2} T^{21} + 4628758875972824499 p^{3} T^{22} + 368691409348339198 p^{4} T^{23} + 49778932803940384 p^{5} T^{24} + 3127977931159775 p^{6} T^{25} + 459400689288747 p^{7} T^{26} + 21038930940961 p^{8} T^{27} + 3558817249441 p^{9} T^{28} + 108208741832 p^{10} T^{29} + 22430153668 p^{11} T^{30} + 400348787 p^{12} T^{31} + 110022826 p^{13} T^{32} + 949536 p^{14} T^{33} + 392410 p^{15} T^{34} + 1085 p^{16} T^{35} + 902 p^{17} T^{36} + p^{19} T^{38} \)
73 \( 1 + 20 T + 1035 T^{2} + 17620 T^{3} + 517098 T^{4} + 7690145 T^{5} + 166572834 T^{6} + 2205603797 T^{7} + 38957981169 T^{8} + 465597466491 T^{9} + 7053722838370 T^{10} + 76835565092557 T^{11} + 1028265501509793 T^{12} + 10278623906882763 T^{13} + 123784113041917842 T^{14} + 1140483585764195683 T^{15} + 12509408888531126726 T^{16} + \)\(10\!\cdots\!53\)\( T^{17} + \)\(10\!\cdots\!84\)\( T^{18} + \)\(84\!\cdots\!70\)\( T^{19} + \)\(10\!\cdots\!84\)\( p T^{20} + \)\(10\!\cdots\!53\)\( p^{2} T^{21} + 12509408888531126726 p^{3} T^{22} + 1140483585764195683 p^{4} T^{23} + 123784113041917842 p^{5} T^{24} + 10278623906882763 p^{6} T^{25} + 1028265501509793 p^{7} T^{26} + 76835565092557 p^{8} T^{27} + 7053722838370 p^{9} T^{28} + 465597466491 p^{10} T^{29} + 38957981169 p^{11} T^{30} + 2205603797 p^{12} T^{31} + 166572834 p^{13} T^{32} + 7690145 p^{14} T^{33} + 517098 p^{15} T^{34} + 17620 p^{16} T^{35} + 1035 p^{17} T^{36} + 20 p^{18} T^{37} + p^{19} T^{38} \)
79 \( 1 + 38 T + 1823 T^{2} + 49357 T^{3} + 1409588 T^{4} + 30186522 T^{5} + 649197903 T^{6} + 11607780631 T^{7} + 204826709148 T^{8} + 3157990204334 T^{9} + 47727241601601 T^{10} + 647838665381407 T^{11} + 8597864411050786 T^{12} + 104185240649215988 T^{13} + 1233381949003359879 T^{14} + 13466476286165832589 T^{15} + \)\(14\!\cdots\!78\)\( T^{16} + \)\(14\!\cdots\!96\)\( T^{17} + \)\(17\!\cdots\!86\)\( p T^{18} + \)\(12\!\cdots\!36\)\( T^{19} + \)\(17\!\cdots\!86\)\( p^{2} T^{20} + \)\(14\!\cdots\!96\)\( p^{2} T^{21} + \)\(14\!\cdots\!78\)\( p^{3} T^{22} + 13466476286165832589 p^{4} T^{23} + 1233381949003359879 p^{5} T^{24} + 104185240649215988 p^{6} T^{25} + 8597864411050786 p^{7} T^{26} + 647838665381407 p^{8} T^{27} + 47727241601601 p^{9} T^{28} + 3157990204334 p^{10} T^{29} + 204826709148 p^{11} T^{30} + 11607780631 p^{12} T^{31} + 649197903 p^{13} T^{32} + 30186522 p^{14} T^{33} + 1409588 p^{15} T^{34} + 49357 p^{16} T^{35} + 1823 p^{17} T^{36} + 38 p^{18} T^{37} + p^{19} T^{38} \)
83 \( 1 + 36 T + 1320 T^{2} + 31176 T^{3} + 708266 T^{4} + 13095311 T^{5} + 231851717 T^{6} + 3632606314 T^{7} + 54604228491 T^{8} + 756432311401 T^{9} + 10071111067524 T^{10} + 126365511530507 T^{11} + 1525307869152585 T^{12} + 17581725050697915 T^{13} + 195062126290088254 T^{14} + 2083294213860253602 T^{15} + 21423143737594143167 T^{16} + \)\(21\!\cdots\!09\)\( T^{17} + \)\(20\!\cdots\!11\)\( T^{18} + \)\(18\!\cdots\!78\)\( T^{19} + \)\(20\!\cdots\!11\)\( p T^{20} + \)\(21\!\cdots\!09\)\( p^{2} T^{21} + 21423143737594143167 p^{3} T^{22} + 2083294213860253602 p^{4} T^{23} + 195062126290088254 p^{5} T^{24} + 17581725050697915 p^{6} T^{25} + 1525307869152585 p^{7} T^{26} + 126365511530507 p^{8} T^{27} + 10071111067524 p^{9} T^{28} + 756432311401 p^{10} T^{29} + 54604228491 p^{11} T^{30} + 3632606314 p^{12} T^{31} + 231851717 p^{13} T^{32} + 13095311 p^{14} T^{33} + 708266 p^{15} T^{34} + 31176 p^{16} T^{35} + 1320 p^{17} T^{36} + 36 p^{18} T^{37} + p^{19} T^{38} \)
89 \( 1 + 25 T + 1294 T^{2} + 26250 T^{3} + 782226 T^{4} + 13481997 T^{5} + 299641351 T^{6} + 4524428234 T^{7} + 82708614115 T^{8} + 1117468413781 T^{9} + 17657786024838 T^{10} + 216628121395716 T^{11} + 3045563398391669 T^{12} + 34266852135290057 T^{13} + 436381940160735136 T^{14} + 4532093373333428262 T^{15} + 52879632718988757665 T^{16} + \)\(50\!\cdots\!48\)\( T^{17} + \)\(54\!\cdots\!73\)\( T^{18} + \)\(48\!\cdots\!40\)\( T^{19} + \)\(54\!\cdots\!73\)\( p T^{20} + \)\(50\!\cdots\!48\)\( p^{2} T^{21} + 52879632718988757665 p^{3} T^{22} + 4532093373333428262 p^{4} T^{23} + 436381940160735136 p^{5} T^{24} + 34266852135290057 p^{6} T^{25} + 3045563398391669 p^{7} T^{26} + 216628121395716 p^{8} T^{27} + 17657786024838 p^{9} T^{28} + 1117468413781 p^{10} T^{29} + 82708614115 p^{11} T^{30} + 4524428234 p^{12} T^{31} + 299641351 p^{13} T^{32} + 13481997 p^{14} T^{33} + 782226 p^{15} T^{34} + 26250 p^{16} T^{35} + 1294 p^{17} T^{36} + 25 p^{18} T^{37} + p^{19} T^{38} \)
97 \( 1 + 13 T + 922 T^{2} + 10182 T^{3} + 416924 T^{4} + 4067466 T^{5} + 125743955 T^{6} + 1118163431 T^{7} + 28782848457 T^{8} + 239023088765 T^{9} + 5358815293129 T^{10} + 42274589977300 T^{11} + 845335550817640 T^{12} + 6396442512504985 T^{13} + 115884929956805182 T^{14} + 843456646689364365 T^{15} + 14033880049339959377 T^{16} + 97976223132614554835 T^{17} + \)\(15\!\cdots\!17\)\( T^{18} + \)\(10\!\cdots\!80\)\( T^{19} + \)\(15\!\cdots\!17\)\( p T^{20} + 97976223132614554835 p^{2} T^{21} + 14033880049339959377 p^{3} T^{22} + 843456646689364365 p^{4} T^{23} + 115884929956805182 p^{5} T^{24} + 6396442512504985 p^{6} T^{25} + 845335550817640 p^{7} T^{26} + 42274589977300 p^{8} T^{27} + 5358815293129 p^{9} T^{28} + 239023088765 p^{10} T^{29} + 28782848457 p^{11} T^{30} + 1118163431 p^{12} T^{31} + 125743955 p^{13} T^{32} + 4067466 p^{14} T^{33} + 416924 p^{15} T^{34} + 10182 p^{16} T^{35} + 922 p^{17} T^{36} + 13 p^{18} T^{37} + p^{19} T^{38} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{38} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.35774962156931810740124080077, −2.35121928740468420458617227832, −2.33477307279299007602425741658, −2.20287916561794040800393127710, −2.19531279571354279761448785681, −1.95567302675646714990980183304, −1.89447757224690494296071058385, −1.79541523958990391462715464688, −1.78360180928001586801246394992, −1.77232324285410911951801193045, −1.71017924376825611071825026220, −1.69689576641308101280967977533, −1.57461544171146835514338887297, −1.54097321056449223719233099180, −1.49039218470837446447822748832, −1.48011405788428173584104020483, −1.38153870052590982146588171181, −1.35011421379858675837205741532, −1.32304337494271512809359859087, −1.27545318359038631543091908218, −1.14235944175899039301871146016, −1.09639042566798634078494325829, −1.05065915857955406434796422669, −0.893562301369399154648904062134, −0.867455266484209134552466148585, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.867455266484209134552466148585, 0.893562301369399154648904062134, 1.05065915857955406434796422669, 1.09639042566798634078494325829, 1.14235944175899039301871146016, 1.27545318359038631543091908218, 1.32304337494271512809359859087, 1.35011421379858675837205741532, 1.38153870052590982146588171181, 1.48011405788428173584104020483, 1.49039218470837446447822748832, 1.54097321056449223719233099180, 1.57461544171146835514338887297, 1.69689576641308101280967977533, 1.71017924376825611071825026220, 1.77232324285410911951801193045, 1.78360180928001586801246394992, 1.79541523958990391462715464688, 1.89447757224690494296071058385, 1.95567302675646714990980183304, 2.19531279571354279761448785681, 2.20287916561794040800393127710, 2.33477307279299007602425741658, 2.35121928740468420458617227832, 2.35774962156931810740124080077

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.