Properties

Degree 64
Conductor $ 3^{32} \cdot 5^{32} \cdot 7^{64} $
Sign $1$
Motivic weight 1
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·8-s + 16·11-s + 4·16-s + 40·23-s − 32·37-s − 32·43-s − 24·53-s + 128·64-s + 32·67-s + 128·71-s + 4·81-s + 256·88-s + 8·107-s + 16·113-s + 224·121-s + 127-s + 56·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 64·176-s + ⋯
L(s)  = 1  + 5.65·8-s + 4.82·11-s + 16-s + 8.34·23-s − 5.26·37-s − 4.87·43-s − 3.29·53-s + 16·64-s + 3.90·67-s + 15.1·71-s + 4/9·81-s + 27.2·88-s + 0.773·107-s + 1.50·113-s + 20.3·121-s + 0.0887·127-s + 4.94·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 4.82·176-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{32} \cdot 7^{64}\right)^{s/2} \, \Gamma_{\C}(s)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{32} \cdot 7^{64}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{32} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(64\)
\( N \)  =  \(3^{32} \cdot 5^{32} \cdot 7^{64}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  induced by $\chi_{735} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((64,\ 3^{32} \cdot 5^{32} \cdot 7^{64} ,\ ( \ : [1/2]^{32} ),\ 1 )\)
\(L(1)\)  \(\approx\)  \(29.3295\)
\(L(\frac12)\)  \(\approx\)  \(29.3295\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 64. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 63.
$p$$F_p(T)$
bad3 \( ( 1 - T^{4} + T^{8} )^{4} \)
5 \( 1 - 28 T^{4} - 512 T^{6} + 162 p T^{8} + 13568 T^{10} + 101264 T^{12} - 365056 T^{14} - 2518349 T^{16} - 365056 p^{2} T^{18} + 101264 p^{4} T^{20} + 13568 p^{6} T^{22} + 162 p^{9} T^{24} - 512 p^{10} T^{26} - 28 p^{12} T^{28} + p^{16} T^{32} \)
7 \( 1 \)
good2 \( ( 1 - p T + p^{2} T^{3} - 5 T^{4} + p^{3} T^{6} - 3 p T^{7} - 7 T^{8} - 3 p^{2} T^{9} + p^{5} T^{10} - 5 p^{4} T^{12} + p^{7} T^{13} - p^{8} T^{15} + p^{8} T^{16} )^{2}( 1 + p T + p^{2} T^{2} - p^{2} T^{3} - 13 T^{4} - p^{5} T^{5} - p^{2} T^{6} + 17 p T^{7} + 121 T^{8} + 17 p^{2} T^{9} - p^{4} T^{10} - p^{8} T^{11} - 13 p^{4} T^{12} - p^{7} T^{13} + p^{8} T^{14} + p^{8} T^{15} + p^{8} T^{16} )^{2} \)
11 \( ( 1 - 4 T - 16 T^{2} + 8 T^{3} + 350 T^{4} + 292 T^{5} - 3488 T^{6} - 276 T^{7} + 14787 T^{8} - 276 p T^{9} - 3488 p^{2} T^{10} + 292 p^{3} T^{11} + 350 p^{4} T^{12} + 8 p^{5} T^{13} - 16 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{4} \)
13 \( ( 1 + 424 T^{4} + 47004 T^{8} - 12160488 T^{12} - 4129271418 T^{16} - 12160488 p^{4} T^{20} + 47004 p^{8} T^{24} + 424 p^{12} T^{28} + p^{16} T^{32} )^{2} \)
17 \( 1 - 120 T^{4} - 152156 T^{8} - 39238288 T^{12} + 16658124042 T^{16} + 314369192184 p T^{20} + 474547971699984 T^{24} - 359345167779328392 T^{28} - 83368676282420650157 T^{32} - 359345167779328392 p^{4} T^{36} + 474547971699984 p^{8} T^{40} + 314369192184 p^{13} T^{44} + 16658124042 p^{16} T^{48} - 39238288 p^{20} T^{52} - 152156 p^{24} T^{56} - 120 p^{28} T^{60} + p^{32} T^{64} \)
19 \( ( 1 - 48 T^{2} + 780 T^{4} + 6880 T^{6} - 416438 T^{8} + 3121456 T^{10} + 58094512 T^{12} - 503650768 T^{14} - 11412226189 T^{16} - 503650768 p^{2} T^{18} + 58094512 p^{4} T^{20} + 3121456 p^{6} T^{22} - 416438 p^{8} T^{24} + 6880 p^{10} T^{26} + 780 p^{12} T^{28} - 48 p^{14} T^{30} + p^{16} T^{32} )^{2} \)
23 \( ( 1 - 20 T + 200 T^{2} - 968 T^{3} - 708 T^{4} + 46980 T^{5} - 329488 T^{6} + 32372 p T^{7} + 5716730 T^{8} - 62265808 T^{9} + 251522648 T^{10} + 104430132 T^{11} - 7339405936 T^{12} + 42204492828 T^{13} - 63690909384 T^{14} - 675893753360 T^{15} + 5483394669219 T^{16} - 675893753360 p T^{17} - 63690909384 p^{2} T^{18} + 42204492828 p^{3} T^{19} - 7339405936 p^{4} T^{20} + 104430132 p^{5} T^{21} + 251522648 p^{6} T^{22} - 62265808 p^{7} T^{23} + 5716730 p^{8} T^{24} + 32372 p^{10} T^{25} - 329488 p^{10} T^{26} + 46980 p^{11} T^{27} - 708 p^{12} T^{28} - 968 p^{13} T^{29} + 200 p^{14} T^{30} - 20 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
29 \( ( 1 - 184 T^{2} + 15868 T^{4} - 835400 T^{6} + 29324070 T^{8} - 835400 p^{2} T^{10} + 15868 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} )^{4} \)
31 \( ( 1 + 128 T^{2} + 6988 T^{4} + 276608 T^{6} + 12312298 T^{8} + 16762432 p T^{10} + 17858096304 T^{12} + 596558285056 T^{14} + 19655246793235 T^{16} + 596558285056 p^{2} T^{18} + 17858096304 p^{4} T^{20} + 16762432 p^{7} T^{22} + 12312298 p^{8} T^{24} + 276608 p^{10} T^{26} + 6988 p^{12} T^{28} + 128 p^{14} T^{30} + p^{16} T^{32} )^{2} \)
37 \( ( 1 + 16 T + 128 T^{2} + 160 T^{3} - 7580 T^{4} - 89136 T^{5} - 443136 T^{6} + 522928 T^{7} + 27958378 T^{8} + 224581824 T^{9} + 711947648 T^{10} - 3897361488 T^{11} - 64328148336 T^{12} - 373358607184 T^{13} - 458984218240 T^{14} + 10863004394560 T^{15} + 104432497832371 T^{16} + 10863004394560 p T^{17} - 458984218240 p^{2} T^{18} - 373358607184 p^{3} T^{19} - 64328148336 p^{4} T^{20} - 3897361488 p^{5} T^{21} + 711947648 p^{6} T^{22} + 224581824 p^{7} T^{23} + 27958378 p^{8} T^{24} + 522928 p^{9} T^{25} - 443136 p^{10} T^{26} - 89136 p^{11} T^{27} - 7580 p^{12} T^{28} + 160 p^{13} T^{29} + 128 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
41 \( ( 1 - 144 T^{2} + 316 p T^{4} - 823792 T^{6} + 38320198 T^{8} - 823792 p^{2} T^{10} + 316 p^{5} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} )^{4} \)
43 \( ( 1 + 8 T + 32 T^{2} + 280 T^{3} + 2788 T^{4} + 14232 T^{5} + 63840 T^{6} + 569416 T^{7} + 5017638 T^{8} + 569416 p T^{9} + 63840 p^{2} T^{10} + 14232 p^{3} T^{11} + 2788 p^{4} T^{12} + 280 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{4} \)
47 \( 1 - 3784 T^{4} + 13193892 T^{8} + 14019931408 T^{12} - 96851625761014 T^{16} + 425990295262260936 T^{20} - \)\(10\!\cdots\!44\)\( T^{24} - \)\(10\!\cdots\!68\)\( T^{28} + \)\(63\!\cdots\!55\)\( T^{32} - \)\(10\!\cdots\!68\)\( p^{4} T^{36} - \)\(10\!\cdots\!44\)\( p^{8} T^{40} + 425990295262260936 p^{12} T^{44} - 96851625761014 p^{16} T^{48} + 14019931408 p^{20} T^{52} + 13193892 p^{24} T^{56} - 3784 p^{28} T^{60} + p^{32} T^{64} \)
53 \( ( 1 + 12 T + 72 T^{2} - 280 T^{3} - 3460 T^{4} - 2972 T^{5} + 252656 T^{6} + 1884420 T^{7} + 7010618 T^{8} + 36681712 T^{9} + 286384600 T^{10} + 2223651740 T^{11} + 43549407952 T^{12} + 535378589820 T^{13} + 3552480552824 T^{14} - 6433766477552 T^{15} - 134952578232669 T^{16} - 6433766477552 p T^{17} + 3552480552824 p^{2} T^{18} + 535378589820 p^{3} T^{19} + 43549407952 p^{4} T^{20} + 2223651740 p^{5} T^{21} + 286384600 p^{6} T^{22} + 36681712 p^{7} T^{23} + 7010618 p^{8} T^{24} + 1884420 p^{9} T^{25} + 252656 p^{10} T^{26} - 2972 p^{11} T^{27} - 3460 p^{12} T^{28} - 280 p^{13} T^{29} + 72 p^{14} T^{30} + 12 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
59 \( ( 1 - 312 T^{2} + 47684 T^{4} - 5412496 T^{6} + 539236842 T^{8} - 46862269256 T^{10} + 3549286946960 T^{12} - 243467978111240 T^{14} + 15175519845036019 T^{16} - 243467978111240 p^{2} T^{18} + 3549286946960 p^{4} T^{20} - 46862269256 p^{6} T^{22} + 539236842 p^{8} T^{24} - 5412496 p^{10} T^{26} + 47684 p^{12} T^{28} - 312 p^{14} T^{30} + p^{16} T^{32} )^{2} \)
61 \( ( 1 + 200 T^{2} + 22468 T^{4} + 1790576 T^{6} + 97855786 T^{8} + 3060068536 T^{10} - 81993057648 T^{12} - 20538307246472 T^{14} - 1625691863741645 T^{16} - 20538307246472 p^{2} T^{18} - 81993057648 p^{4} T^{20} + 3060068536 p^{6} T^{22} + 97855786 p^{8} T^{24} + 1790576 p^{10} T^{26} + 22468 p^{12} T^{28} + 200 p^{14} T^{30} + p^{16} T^{32} )^{2} \)
67 \( ( 1 - 16 T + 128 T^{2} + 800 T^{3} - 28836 T^{4} + 324400 T^{5} - 1179392 T^{6} - 18092176 T^{7} + 5252238 p T^{8} - 2925843392 T^{9} + 5905292672 T^{10} + 176583458736 T^{11} - 2663667650960 T^{12} + 18475343271952 T^{13} - 13556445594240 T^{14} - 1177472024707520 T^{15} + 14449494060791283 T^{16} - 1177472024707520 p T^{17} - 13556445594240 p^{2} T^{18} + 18475343271952 p^{3} T^{19} - 2663667650960 p^{4} T^{20} + 176583458736 p^{5} T^{21} + 5905292672 p^{6} T^{22} - 2925843392 p^{7} T^{23} + 5252238 p^{9} T^{24} - 18092176 p^{9} T^{25} - 1179392 p^{10} T^{26} + 324400 p^{11} T^{27} - 28836 p^{12} T^{28} + 800 p^{13} T^{29} + 128 p^{14} T^{30} - 16 p^{15} T^{31} + p^{16} T^{32} )^{2} \)
71 \( ( 1 - 16 T + 232 T^{2} - 2376 T^{3} + 21730 T^{4} - 2376 p T^{5} + 232 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} )^{8} \)
73 \( 1 + 15256 T^{4} + 151883236 T^{8} + 1421742645328 T^{12} + 10335613149556490 T^{16} + 61793546475337843048 T^{20} + \)\(37\!\cdots\!40\)\( T^{24} + \)\(19\!\cdots\!60\)\( T^{28} + \)\(10\!\cdots\!51\)\( T^{32} + \)\(19\!\cdots\!60\)\( p^{4} T^{36} + \)\(37\!\cdots\!40\)\( p^{8} T^{40} + 61793546475337843048 p^{12} T^{44} + 10335613149556490 p^{16} T^{48} + 1421742645328 p^{20} T^{52} + 151883236 p^{24} T^{56} + 15256 p^{28} T^{60} + p^{32} T^{64} \)
79 \( ( 1 + 312 T^{2} + 38948 T^{4} + 3632912 T^{6} + 462435210 T^{8} + 52001862856 T^{10} + 4289689557776 T^{12} + 371548575974344 T^{14} + 32785767691048147 T^{16} + 371548575974344 p^{2} T^{18} + 4289689557776 p^{4} T^{20} + 52001862856 p^{6} T^{22} + 462435210 p^{8} T^{24} + 3632912 p^{10} T^{26} + 38948 p^{12} T^{28} + 312 p^{14} T^{30} + p^{16} T^{32} )^{2} \)
83 \( ( 1 + 5000 T^{4} + 95818588 T^{8} + 596752860728 T^{12} + 4571727903671302 T^{16} + 596752860728 p^{4} T^{20} + 95818588 p^{8} T^{24} + 5000 p^{12} T^{28} + p^{16} T^{32} )^{2} \)
89 \( ( 1 - 576 T^{2} + 176708 T^{4} - 38642560 T^{6} + 6710843274 T^{8} - 972949091264 T^{10} + 120937392753680 T^{12} - 13081128365249984 T^{14} + 1241287180885474387 T^{16} - 13081128365249984 p^{2} T^{18} + 120937392753680 p^{4} T^{20} - 972949091264 p^{6} T^{22} + 6710843274 p^{8} T^{24} - 38642560 p^{10} T^{26} + 176708 p^{12} T^{28} - 576 p^{14} T^{30} + p^{16} T^{32} )^{2} \)
97 \( ( 1 - 55064 T^{4} + 1465436892 T^{8} - 24336256217256 T^{12} + 274732504520067270 T^{16} - 24336256217256 p^{4} T^{20} + 1465436892 p^{8} T^{24} - 55064 p^{12} T^{28} + p^{16} T^{32} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{64} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−1.58715963260622480740015380394, −1.56819394968077822129608119758, −1.47907196446438857129333517165, −1.47889903297696870742016281359, −1.47216825342047502314541192110, −1.46169110961558286673290034078, −1.42904997497532151648415943705, −1.33883398695939114863828061841, −1.27788593587978143864464687669, −1.26040628810906108659559384327, −1.19783250988115545206397730368, −1.14791690118551016948437351597, −1.14626395586208455604178708357, −1.13598076038366903758862658819, −0.938763890896463847573910205088, −0.798828188844792395196980466776, −0.76689644521248334070844293384, −0.74721852781424225259354661273, −0.71072741413049919656289534233, −0.67675534729583733062235122390, −0.59728747481386606212696251923, −0.59627783858456104380322476066, −0.40744450130136418879187670404, −0.35589677643201921255165083199, −0.02283995737772526677625638078, 0.02283995737772526677625638078, 0.35589677643201921255165083199, 0.40744450130136418879187670404, 0.59627783858456104380322476066, 0.59728747481386606212696251923, 0.67675534729583733062235122390, 0.71072741413049919656289534233, 0.74721852781424225259354661273, 0.76689644521248334070844293384, 0.798828188844792395196980466776, 0.938763890896463847573910205088, 1.13598076038366903758862658819, 1.14626395586208455604178708357, 1.14791690118551016948437351597, 1.19783250988115545206397730368, 1.26040628810906108659559384327, 1.27788593587978143864464687669, 1.33883398695939114863828061841, 1.42904997497532151648415943705, 1.46169110961558286673290034078, 1.47216825342047502314541192110, 1.47889903297696870742016281359, 1.47907196446438857129333517165, 1.56819394968077822129608119758, 1.58715963260622480740015380394

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

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