Properties

Degree 28
Conductor $ 3^{14} \cdot 7^{28} \cdot 41^{14} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 14

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 14·3-s − 5·4-s − 10·5-s − 28·6-s + 12·8-s + 105·9-s + 20·10-s − 16·11-s − 70·12-s − 21·13-s − 140·15-s + 11·16-s − 12·17-s − 210·18-s − 2·19-s + 50·20-s + 32·22-s − 7·23-s + 168·24-s + 26·25-s + 42·26-s + 560·27-s − 16·29-s + 280·30-s − 8·31-s − 35·32-s + ⋯
L(s)  = 1  − 1.41·2-s + 8.08·3-s − 5/2·4-s − 4.47·5-s − 11.4·6-s + 4.24·8-s + 35·9-s + 6.32·10-s − 4.82·11-s − 20.2·12-s − 5.82·13-s − 36.1·15-s + 11/4·16-s − 2.91·17-s − 49.4·18-s − 0.458·19-s + 11.1·20-s + 6.82·22-s − 1.45·23-s + 34.2·24-s + 26/5·25-s + 8.23·26-s + 107.·27-s − 2.97·29-s + 51.1·30-s − 1.43·31-s − 6.18·32-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{14} \cdot 7^{28} \cdot 41^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr =\mathstrut & \,\Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(3^{14} \cdot 7^{28} \cdot 41^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(28\)
\( N \)  =  \(3^{14} \cdot 7^{28} \cdot 41^{14}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{6027} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  14
Selberg data  =  $(28,\ 3^{14} \cdot 7^{28} \cdot 41^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;41\}$, \(F_p\) is a polynomial of degree 28. If $p \in \{3,\;7,\;41\}$, then $F_p$ is a polynomial of degree at most 27.
$p$$F_p$
bad3 \( ( 1 - T )^{14} \)
7 \( 1 \)
41 \( ( 1 + T )^{14} \)
good2 \( 1 + p T + 9 T^{2} + p^{4} T^{3} + 21 p T^{4} + 69 T^{5} + 69 p T^{6} + 103 p T^{7} + 175 p T^{8} + 471 T^{9} + 91 p^{3} T^{10} + 899 T^{11} + 341 p^{2} T^{12} + 827 p T^{13} + 41 p^{6} T^{14} + 827 p^{2} T^{15} + 341 p^{4} T^{16} + 899 p^{3} T^{17} + 91 p^{7} T^{18} + 471 p^{5} T^{19} + 175 p^{7} T^{20} + 103 p^{8} T^{21} + 69 p^{9} T^{22} + 69 p^{9} T^{23} + 21 p^{11} T^{24} + p^{15} T^{25} + 9 p^{12} T^{26} + p^{14} T^{27} + p^{14} T^{28} \)
5 \( 1 + 2 p T + 74 T^{2} + 397 T^{3} + 362 p T^{4} + 1412 p T^{5} + 24952 T^{6} + 80389 T^{7} + 243041 T^{8} + 137996 p T^{9} + 373087 p T^{10} + 4793748 T^{11} + 11815128 T^{12} + 27903508 T^{13} + 63605864 T^{14} + 27903508 p T^{15} + 11815128 p^{2} T^{16} + 4793748 p^{3} T^{17} + 373087 p^{5} T^{18} + 137996 p^{6} T^{19} + 243041 p^{6} T^{20} + 80389 p^{7} T^{21} + 24952 p^{8} T^{22} + 1412 p^{10} T^{23} + 362 p^{11} T^{24} + 397 p^{11} T^{25} + 74 p^{12} T^{26} + 2 p^{14} T^{27} + p^{14} T^{28} \)
11 \( 1 + 16 T + 184 T^{2} + 1575 T^{3} + 11360 T^{4} + 70325 T^{5} + 388857 T^{6} + 1938645 T^{7} + 8900582 T^{8} + 37871442 T^{9} + 151561637 T^{10} + 573710944 T^{11} + 2078763437 T^{12} + 7237578515 T^{13} + 24407865486 T^{14} + 7237578515 p T^{15} + 2078763437 p^{2} T^{16} + 573710944 p^{3} T^{17} + 151561637 p^{4} T^{18} + 37871442 p^{5} T^{19} + 8900582 p^{6} T^{20} + 1938645 p^{7} T^{21} + 388857 p^{8} T^{22} + 70325 p^{9} T^{23} + 11360 p^{10} T^{24} + 1575 p^{11} T^{25} + 184 p^{12} T^{26} + 16 p^{13} T^{27} + p^{14} T^{28} \)
13 \( 1 + 21 T + 271 T^{2} + 2413 T^{3} + 16847 T^{4} + 7273 p T^{5} + 450699 T^{6} + 1857513 T^{7} + 7040959 T^{8} + 25462284 T^{9} + 95065429 T^{10} + 365768395 T^{11} + 1458818515 T^{12} + 433746645 p T^{13} + 21060922205 T^{14} + 433746645 p^{2} T^{15} + 1458818515 p^{2} T^{16} + 365768395 p^{3} T^{17} + 95065429 p^{4} T^{18} + 25462284 p^{5} T^{19} + 7040959 p^{6} T^{20} + 1857513 p^{7} T^{21} + 450699 p^{8} T^{22} + 7273 p^{10} T^{23} + 16847 p^{10} T^{24} + 2413 p^{11} T^{25} + 271 p^{12} T^{26} + 21 p^{13} T^{27} + p^{14} T^{28} \)
17 \( 1 + 12 T + 199 T^{2} + 1849 T^{3} + 18186 T^{4} + 138947 T^{5} + 1038737 T^{6} + 6762833 T^{7} + 42011572 T^{8} + 238366545 T^{9} + 1282718820 T^{10} + 6434014451 T^{11} + 30621499975 T^{12} + 136944093771 T^{13} + 581869551548 T^{14} + 136944093771 p T^{15} + 30621499975 p^{2} T^{16} + 6434014451 p^{3} T^{17} + 1282718820 p^{4} T^{18} + 238366545 p^{5} T^{19} + 42011572 p^{6} T^{20} + 6762833 p^{7} T^{21} + 1038737 p^{8} T^{22} + 138947 p^{9} T^{23} + 18186 p^{10} T^{24} + 1849 p^{11} T^{25} + 199 p^{12} T^{26} + 12 p^{13} T^{27} + p^{14} T^{28} \)
19 \( 1 + 2 T + 103 T^{2} + 200 T^{3} + 5623 T^{4} + 9811 T^{5} + 218060 T^{6} + 17526 p T^{7} + 6745499 T^{8} + 8994164 T^{9} + 176627464 T^{10} + 206515868 T^{11} + 4025743533 T^{12} + 4244124076 T^{13} + 81094566611 T^{14} + 4244124076 p T^{15} + 4025743533 p^{2} T^{16} + 206515868 p^{3} T^{17} + 176627464 p^{4} T^{18} + 8994164 p^{5} T^{19} + 6745499 p^{6} T^{20} + 17526 p^{8} T^{21} + 218060 p^{8} T^{22} + 9811 p^{9} T^{23} + 5623 p^{10} T^{24} + 200 p^{11} T^{25} + 103 p^{12} T^{26} + 2 p^{13} T^{27} + p^{14} T^{28} \)
23 \( 1 + 7 T + 168 T^{2} + 1074 T^{3} + 14546 T^{4} + 87731 T^{5} + 856828 T^{6} + 4935839 T^{7} + 38309166 T^{8} + 211173540 T^{9} + 1375595290 T^{10} + 7207565030 T^{11} + 40911554559 T^{12} + 200940540495 T^{13} + 1023524456646 T^{14} + 200940540495 p T^{15} + 40911554559 p^{2} T^{16} + 7207565030 p^{3} T^{17} + 1375595290 p^{4} T^{18} + 211173540 p^{5} T^{19} + 38309166 p^{6} T^{20} + 4935839 p^{7} T^{21} + 856828 p^{8} T^{22} + 87731 p^{9} T^{23} + 14546 p^{10} T^{24} + 1074 p^{11} T^{25} + 168 p^{12} T^{26} + 7 p^{13} T^{27} + p^{14} T^{28} \)
29 \( 1 + 16 T + 317 T^{2} + 3532 T^{3} + 41506 T^{4} + 358542 T^{5} + 3166373 T^{6} + 22457151 T^{7} + 163208335 T^{8} + 988563189 T^{9} + 6264006001 T^{10} + 1164934708 p T^{11} + 197938095774 T^{12} + 1007044565606 T^{13} + 5786350097514 T^{14} + 1007044565606 p T^{15} + 197938095774 p^{2} T^{16} + 1164934708 p^{4} T^{17} + 6264006001 p^{4} T^{18} + 988563189 p^{5} T^{19} + 163208335 p^{6} T^{20} + 22457151 p^{7} T^{21} + 3166373 p^{8} T^{22} + 358542 p^{9} T^{23} + 41506 p^{10} T^{24} + 3532 p^{11} T^{25} + 317 p^{12} T^{26} + 16 p^{13} T^{27} + p^{14} T^{28} \)
31 \( 1 + 8 T + 222 T^{2} + 1560 T^{3} + 829 p T^{4} + 166626 T^{5} + 2077439 T^{6} + 12462414 T^{7} + 128404744 T^{8} + 712573054 T^{9} + 6362583331 T^{10} + 32639362525 T^{11} + 259228460931 T^{12} + 1223573010128 T^{13} + 8789060708753 T^{14} + 1223573010128 p T^{15} + 259228460931 p^{2} T^{16} + 32639362525 p^{3} T^{17} + 6362583331 p^{4} T^{18} + 712573054 p^{5} T^{19} + 128404744 p^{6} T^{20} + 12462414 p^{7} T^{21} + 2077439 p^{8} T^{22} + 166626 p^{9} T^{23} + 829 p^{11} T^{24} + 1560 p^{11} T^{25} + 222 p^{12} T^{26} + 8 p^{13} T^{27} + p^{14} T^{28} \)
37 \( 1 - T + 183 T^{2} - 12 T^{3} + 18163 T^{4} + 15928 T^{5} + 1328947 T^{6} + 2069910 T^{7} + 79538603 T^{8} + 156366354 T^{9} + 4054280539 T^{10} + 8925367239 T^{11} + 179665035801 T^{12} + 407879067683 T^{13} + 7040743815363 T^{14} + 407879067683 p T^{15} + 179665035801 p^{2} T^{16} + 8925367239 p^{3} T^{17} + 4054280539 p^{4} T^{18} + 156366354 p^{5} T^{19} + 79538603 p^{6} T^{20} + 2069910 p^{7} T^{21} + 1328947 p^{8} T^{22} + 15928 p^{9} T^{23} + 18163 p^{10} T^{24} - 12 p^{11} T^{25} + 183 p^{12} T^{26} - p^{13} T^{27} + p^{14} T^{28} \)
43 \( 1 - 14 T + 402 T^{2} - 4820 T^{3} + 81572 T^{4} - 850123 T^{5} + 10845105 T^{6} - 99803305 T^{7} + 1051064487 T^{8} - 8635712784 T^{9} + 78419153734 T^{10} - 579251780009 T^{11} + 4644399472968 T^{12} - 30928648501325 T^{13} + 221828973619470 T^{14} - 30928648501325 p T^{15} + 4644399472968 p^{2} T^{16} - 579251780009 p^{3} T^{17} + 78419153734 p^{4} T^{18} - 8635712784 p^{5} T^{19} + 1051064487 p^{6} T^{20} - 99803305 p^{7} T^{21} + 10845105 p^{8} T^{22} - 850123 p^{9} T^{23} + 81572 p^{10} T^{24} - 4820 p^{11} T^{25} + 402 p^{12} T^{26} - 14 p^{13} T^{27} + p^{14} T^{28} \)
47 \( 1 + 12 T + 465 T^{2} + 4231 T^{3} + 94682 T^{4} + 679890 T^{5} + 11767845 T^{6} + 69149233 T^{7} + 1055970810 T^{8} + 5312440479 T^{9} + 76051474801 T^{10} + 342534768515 T^{11} + 4613008886395 T^{12} + 19010104796746 T^{13} + 236467153328632 T^{14} + 19010104796746 p T^{15} + 4613008886395 p^{2} T^{16} + 342534768515 p^{3} T^{17} + 76051474801 p^{4} T^{18} + 5312440479 p^{5} T^{19} + 1055970810 p^{6} T^{20} + 69149233 p^{7} T^{21} + 11767845 p^{8} T^{22} + 679890 p^{9} T^{23} + 94682 p^{10} T^{24} + 4231 p^{11} T^{25} + 465 p^{12} T^{26} + 12 p^{13} T^{27} + p^{14} T^{28} \)
53 \( 1 + 20 T + 708 T^{2} + 11302 T^{3} + 226311 T^{4} + 3014721 T^{5} + 44161334 T^{6} + 504792523 T^{7} + 5955155121 T^{8} + 59473293082 T^{9} + 592795568408 T^{10} + 5229964592709 T^{11} + 45244413374513 T^{12} + 354465691127099 T^{13} + 2702094398926392 T^{14} + 354465691127099 p T^{15} + 45244413374513 p^{2} T^{16} + 5229964592709 p^{3} T^{17} + 592795568408 p^{4} T^{18} + 59473293082 p^{5} T^{19} + 5955155121 p^{6} T^{20} + 504792523 p^{7} T^{21} + 44161334 p^{8} T^{22} + 3014721 p^{9} T^{23} + 226311 p^{10} T^{24} + 11302 p^{11} T^{25} + 708 p^{12} T^{26} + 20 p^{13} T^{27} + p^{14} T^{28} \)
59 \( 1 + 25 T + 761 T^{2} + 13590 T^{3} + 249489 T^{4} + 3553596 T^{5} + 49726898 T^{6} + 597551814 T^{7} + 6938832060 T^{8} + 72615000269 T^{9} + 729075201514 T^{10} + 6765227449729 T^{11} + 60019732183268 T^{12} + 498276090891047 T^{13} + 3949081180758810 T^{14} + 498276090891047 p T^{15} + 60019732183268 p^{2} T^{16} + 6765227449729 p^{3} T^{17} + 729075201514 p^{4} T^{18} + 72615000269 p^{5} T^{19} + 6938832060 p^{6} T^{20} + 597551814 p^{7} T^{21} + 49726898 p^{8} T^{22} + 3553596 p^{9} T^{23} + 249489 p^{10} T^{24} + 13590 p^{11} T^{25} + 761 p^{12} T^{26} + 25 p^{13} T^{27} + p^{14} T^{28} \)
61 \( 1 + 26 T + 724 T^{2} + 12858 T^{3} + 227681 T^{4} + 3231501 T^{5} + 44968054 T^{6} + 541022108 T^{7} + 6352446641 T^{8} + 66760232128 T^{9} + 683873438158 T^{10} + 6380791731123 T^{11} + 58012873940203 T^{12} + 484514582715328 T^{13} + 3943199649863728 T^{14} + 484514582715328 p T^{15} + 58012873940203 p^{2} T^{16} + 6380791731123 p^{3} T^{17} + 683873438158 p^{4} T^{18} + 66760232128 p^{5} T^{19} + 6352446641 p^{6} T^{20} + 541022108 p^{7} T^{21} + 44968054 p^{8} T^{22} + 3231501 p^{9} T^{23} + 227681 p^{10} T^{24} + 12858 p^{11} T^{25} + 724 p^{12} T^{26} + 26 p^{13} T^{27} + p^{14} T^{28} \)
67 \( 1 + 22 T + 786 T^{2} + 12292 T^{3} + 252223 T^{4} + 3063502 T^{5} + 46784121 T^{6} + 469037360 T^{7} + 5994162460 T^{8} + 52497269216 T^{9} + 603628342905 T^{10} + 4830825339343 T^{11} + 51444193281073 T^{12} + 381263974030052 T^{13} + 3745272478086559 T^{14} + 381263974030052 p T^{15} + 51444193281073 p^{2} T^{16} + 4830825339343 p^{3} T^{17} + 603628342905 p^{4} T^{18} + 52497269216 p^{5} T^{19} + 5994162460 p^{6} T^{20} + 469037360 p^{7} T^{21} + 46784121 p^{8} T^{22} + 3063502 p^{9} T^{23} + 252223 p^{10} T^{24} + 12292 p^{11} T^{25} + 786 p^{12} T^{26} + 22 p^{13} T^{27} + p^{14} T^{28} \)
71 \( 1 + 36 T + 1122 T^{2} + 23675 T^{3} + 439683 T^{4} + 6676458 T^{5} + 91075373 T^{6} + 1078715205 T^{7} + 11655221821 T^{8} + 112521867544 T^{9} + 1009334210852 T^{10} + 8327034190240 T^{11} + 66697799360255 T^{12} + 523118452363710 T^{13} + 4326602007607098 T^{14} + 523118452363710 p T^{15} + 66697799360255 p^{2} T^{16} + 8327034190240 p^{3} T^{17} + 1009334210852 p^{4} T^{18} + 112521867544 p^{5} T^{19} + 11655221821 p^{6} T^{20} + 1078715205 p^{7} T^{21} + 91075373 p^{8} T^{22} + 6676458 p^{9} T^{23} + 439683 p^{10} T^{24} + 23675 p^{11} T^{25} + 1122 p^{12} T^{26} + 36 p^{13} T^{27} + p^{14} T^{28} \)
73 \( 1 + 31 T + 1023 T^{2} + 21898 T^{3} + 447401 T^{4} + 7499375 T^{5} + 118438431 T^{6} + 1651224918 T^{7} + 21743332764 T^{8} + 260615021052 T^{9} + 2962223373867 T^{10} + 31105248627116 T^{11} + 310612377724985 T^{12} + 2886053032824887 T^{13} + 25535627088333071 T^{14} + 2886053032824887 p T^{15} + 310612377724985 p^{2} T^{16} + 31105248627116 p^{3} T^{17} + 2962223373867 p^{4} T^{18} + 260615021052 p^{5} T^{19} + 21743332764 p^{6} T^{20} + 1651224918 p^{7} T^{21} + 118438431 p^{8} T^{22} + 7499375 p^{9} T^{23} + 447401 p^{10} T^{24} + 21898 p^{11} T^{25} + 1023 p^{12} T^{26} + 31 p^{13} T^{27} + p^{14} T^{28} \)
79 \( 1 - 12 T + 454 T^{2} - 5128 T^{3} + 115151 T^{4} - 1225079 T^{5} + 20527605 T^{6} - 201709390 T^{7} + 2821623788 T^{8} - 25680305127 T^{9} + 317937143449 T^{10} - 2681951227239 T^{11} + 30479224526413 T^{12} - 240756686983991 T^{13} + 2562192972209775 T^{14} - 240756686983991 p T^{15} + 30479224526413 p^{2} T^{16} - 2681951227239 p^{3} T^{17} + 317937143449 p^{4} T^{18} - 25680305127 p^{5} T^{19} + 2821623788 p^{6} T^{20} - 201709390 p^{7} T^{21} + 20527605 p^{8} T^{22} - 1225079 p^{9} T^{23} + 115151 p^{10} T^{24} - 5128 p^{11} T^{25} + 454 p^{12} T^{26} - 12 p^{13} T^{27} + p^{14} T^{28} \)
83 \( 1 + 20 T + 828 T^{2} + 12456 T^{3} + 293841 T^{4} + 3503139 T^{5} + 60897962 T^{6} + 586694915 T^{7} + 8351045943 T^{8} + 64808597946 T^{9} + 818558637650 T^{10} + 5067579467421 T^{11} + 63887743351779 T^{12} + 336170148000773 T^{13} + 4924830707316186 T^{14} + 336170148000773 p T^{15} + 63887743351779 p^{2} T^{16} + 5067579467421 p^{3} T^{17} + 818558637650 p^{4} T^{18} + 64808597946 p^{5} T^{19} + 8351045943 p^{6} T^{20} + 586694915 p^{7} T^{21} + 60897962 p^{8} T^{22} + 3503139 p^{9} T^{23} + 293841 p^{10} T^{24} + 12456 p^{11} T^{25} + 828 p^{12} T^{26} + 20 p^{13} T^{27} + p^{14} T^{28} \)
89 \( 1 + 39 T + 1284 T^{2} + 27272 T^{3} + 522890 T^{4} + 7732961 T^{5} + 106187855 T^{6} + 1150435046 T^{7} + 11483084106 T^{8} + 78160027692 T^{9} + 369703213409 T^{10} - 3036301927242 T^{11} - 73766007100741 T^{12} - 1171514974709534 T^{13} - 11473336351497890 T^{14} - 1171514974709534 p T^{15} - 73766007100741 p^{2} T^{16} - 3036301927242 p^{3} T^{17} + 369703213409 p^{4} T^{18} + 78160027692 p^{5} T^{19} + 11483084106 p^{6} T^{20} + 1150435046 p^{7} T^{21} + 106187855 p^{8} T^{22} + 7732961 p^{9} T^{23} + 522890 p^{10} T^{24} + 27272 p^{11} T^{25} + 1284 p^{12} T^{26} + 39 p^{13} T^{27} + p^{14} T^{28} \)
97 \( 1 + 18 T + 1054 T^{2} + 16615 T^{3} + 537178 T^{4} + 7492834 T^{5} + 175167348 T^{6} + 2179534055 T^{7} + 40813546689 T^{8} + 455284157750 T^{9} + 7187904120059 T^{10} + 72018427556922 T^{11} + 986836359870026 T^{12} + 8867518005749374 T^{13} + 107299779883061530 T^{14} + 8867518005749374 p T^{15} + 986836359870026 p^{2} T^{16} + 72018427556922 p^{3} T^{17} + 7187904120059 p^{4} T^{18} + 455284157750 p^{5} T^{19} + 40813546689 p^{6} T^{20} + 2179534055 p^{7} T^{21} + 175167348 p^{8} T^{22} + 7492834 p^{9} T^{23} + 537178 p^{10} T^{24} + 16615 p^{11} T^{25} + 1054 p^{12} T^{26} + 18 p^{13} T^{27} + p^{14} T^{28} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.55421316633116235771750547873, −2.54155001581693189373179491158, −2.52945263807440330663330456446, −2.45711544894388942401829934563, −2.24337426657071445423425871729, −2.20314042896357978202569328232, −2.18859080783189454466413911988, −2.15404130062600625421871918208, −2.10087651922802864112537081414, −2.09032167829964142852757543338, −2.05153728292498832648727417824, −2.01419209629349933652639767899, −1.96453248231044472765949995746, −1.90121731472281249403182227332, −1.56973289587647892906186430572, −1.46786501514113450902943218355, −1.41650765711262244401455228205, −1.39355978312033456804145312181, −1.37043932474548682311062930715, −1.34859254131930283217143271422, −1.19611426158060910040759906020, −1.15705411750442199800909476768, −1.14332759804562915405097110772, −0.935801847183346865681483677671, −0.933126193624121192041351459681, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.933126193624121192041351459681, 0.935801847183346865681483677671, 1.14332759804562915405097110772, 1.15705411750442199800909476768, 1.19611426158060910040759906020, 1.34859254131930283217143271422, 1.37043932474548682311062930715, 1.39355978312033456804145312181, 1.41650765711262244401455228205, 1.46786501514113450902943218355, 1.56973289587647892906186430572, 1.90121731472281249403182227332, 1.96453248231044472765949995746, 2.01419209629349933652639767899, 2.05153728292498832648727417824, 2.09032167829964142852757543338, 2.10087651922802864112537081414, 2.15404130062600625421871918208, 2.18859080783189454466413911988, 2.20314042896357978202569328232, 2.24337426657071445423425871729, 2.45711544894388942401829934563, 2.52945263807440330663330456446, 2.54155001581693189373179491158, 2.55421316633116235771750547873

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

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