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Properties

Label 8-162e4-1.1-c9e4-0-5
Degree $8$
Conductor $688747536$
Sign $1$
Analytic cond. $4.84629\times 10^{7}$
Root an. cond. $9.13432$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·2-s + 256·4-s + 912·5-s − 6.44e3·7-s − 8.19e3·8-s + 2.91e4·10-s + 1.51e4·11-s − 2.89e4·13-s − 2.06e5·14-s − 2.62e5·16-s − 7.99e5·17-s + 1.62e4·19-s + 2.33e5·20-s + 4.84e5·22-s + 2.56e6·23-s + 1.68e6·25-s − 9.25e5·26-s − 1.65e6·28-s + 3.05e6·29-s − 1.67e7·31-s − 2.09e6·32-s − 2.55e7·34-s − 5.88e6·35-s + 6.95e7·37-s + 5.18e5·38-s − 7.47e6·40-s − 2.42e7·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 1/2·4-s + 0.652·5-s − 1.01·7-s − 0.707·8-s + 0.922·10-s + 0.311·11-s − 0.280·13-s − 1.43·14-s − 16-s − 2.32·17-s + 0.0285·19-s + 0.326·20-s + 0.440·22-s + 1.91·23-s + 0.863·25-s − 0.397·26-s − 0.507·28-s + 0.803·29-s − 3.26·31-s − 0.353·32-s − 3.28·34-s − 0.662·35-s + 6.10·37-s + 0.0403·38-s − 0.461·40-s − 1.33·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(4.84629\times 10^{7}\)
Root analytic conductor: \(9.13432\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 9/2, 9/2, 9/2, 9/2 ),\ 1 )\)

Particular Values

\(L(5)\) \(\approx\) \(10.88337761\)
\(L(\frac12)\) \(\approx\) \(10.88337761\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )^{2} \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 - 912 T - 855601 T^{2} + 404728272 p T^{3} - 103682564664 p^{2} T^{4} + 404728272 p^{10} T^{5} - 855601 p^{18} T^{6} - 912 p^{27} T^{7} + p^{36} T^{8} \)
7$D_4\times C_2$ \( 1 + 6448 T + 11146339 T^{2} - 46312160336 p T^{3} - 48575029743848 p^{2} T^{4} - 46312160336 p^{10} T^{5} + 11146339 p^{18} T^{6} + 6448 p^{27} T^{7} + p^{36} T^{8} \)
11$D_4\times C_2$ \( 1 - 15126 T - 661352875 T^{2} + 57868243540506 T^{3} - 5307041222109893796 T^{4} + 57868243540506 p^{9} T^{5} - 661352875 p^{18} T^{6} - 15126 p^{27} T^{7} + p^{36} T^{8} \)
13$D_4\times C_2$ \( 1 + 2224 p T - 19611334538 T^{2} - 1694155271936 p T^{3} + \)\(29\!\cdots\!83\)\( T^{4} - 1694155271936 p^{10} T^{5} - 19611334538 p^{18} T^{6} + 2224 p^{28} T^{7} + p^{36} T^{8} \)
17$D_{4}$ \( ( 1 + 399960 T + 159708648994 T^{2} + 399960 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
19$D_{4}$ \( ( 1 - 8104 T + 62707290162 T^{2} - 8104 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 2563572 T + 1677056566862 T^{2} - 3313518095181322512 T^{3} + \)\(88\!\cdots\!23\)\( T^{4} - 3313518095181322512 p^{9} T^{5} + 1677056566862 p^{18} T^{6} - 2563572 p^{27} T^{7} + p^{36} T^{8} \)
29$D_4\times C_2$ \( 1 - 3059088 T - 21459688231030 T^{2} - 5516807238278175168 T^{3} + \)\(60\!\cdots\!79\)\( T^{4} - 5516807238278175168 p^{9} T^{5} - 21459688231030 p^{18} T^{6} - 3059088 p^{27} T^{7} + p^{36} T^{8} \)
31$D_4\times C_2$ \( 1 + 16787920 T + 158785303873483 T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(69\!\cdots\!48\)\( T^{4} + \)\(11\!\cdots\!00\)\( p^{9} T^{5} + 158785303873483 p^{18} T^{6} + 16787920 p^{27} T^{7} + p^{36} T^{8} \)
37$D_{4}$ \( ( 1 - 34798684 T + 556905085507518 T^{2} - 34798684 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 + 24238968 T - 163876515706654 T^{2} + \)\(23\!\cdots\!08\)\( T^{3} + \)\(26\!\cdots\!99\)\( T^{4} + \)\(23\!\cdots\!08\)\( p^{9} T^{5} - 163876515706654 p^{18} T^{6} + 24238968 p^{27} T^{7} + p^{36} T^{8} \)
43$D_4\times C_2$ \( 1 + 12599680 T - 829505740791986 T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(64\!\cdots\!47\)\( T^{4} - \)\(21\!\cdots\!00\)\( p^{9} T^{5} - 829505740791986 p^{18} T^{6} + 12599680 p^{27} T^{7} + p^{36} T^{8} \)
47$D_4\times C_2$ \( 1 + 19535772 T - 511915408404946 T^{2} - \)\(26\!\cdots\!88\)\( T^{3} - \)\(88\!\cdots\!17\)\( T^{4} - \)\(26\!\cdots\!88\)\( p^{9} T^{5} - 511915408404946 p^{18} T^{6} + 19535772 p^{27} T^{7} + p^{36} T^{8} \)
53$D_{4}$ \( ( 1 + 12534912 T + 2108691984493177 T^{2} + 12534912 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 + 140256744 T - 2436243234512326 T^{2} + \)\(67\!\cdots\!96\)\( T^{3} + \)\(24\!\cdots\!19\)\( T^{4} + \)\(67\!\cdots\!96\)\( p^{9} T^{5} - 2436243234512326 p^{18} T^{6} + 140256744 p^{27} T^{7} + p^{36} T^{8} \)
61$D_4\times C_2$ \( 1 - 169724624 T + 1574903457727750 T^{2} - \)\(65\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!79\)\( T^{4} - \)\(65\!\cdots\!56\)\( p^{9} T^{5} + 1574903457727750 p^{18} T^{6} - 169724624 p^{27} T^{7} + p^{36} T^{8} \)
67$D_4\times C_2$ \( 1 - 72241232 T - 8501511783429026 T^{2} + \)\(29\!\cdots\!08\)\( T^{3} - \)\(69\!\cdots\!37\)\( T^{4} + \)\(29\!\cdots\!08\)\( p^{9} T^{5} - 8501511783429026 p^{18} T^{6} - 72241232 p^{27} T^{7} + p^{36} T^{8} \)
71$D_{4}$ \( ( 1 - 53846280 T + 73069037618267662 T^{2} - 53846280 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
73$D_{4}$ \( ( 1 + 203557694 T + 85276940890015635 T^{2} + 203557694 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 877964864 T + 351746051807337634 T^{2} - \)\(15\!\cdots\!36\)\( T^{3} + \)\(68\!\cdots\!59\)\( T^{4} - \)\(15\!\cdots\!36\)\( p^{9} T^{5} + 351746051807337634 p^{18} T^{6} - 877964864 p^{27} T^{7} + p^{36} T^{8} \)
83$D_4\times C_2$ \( 1 + 111456714 T - 135717001842620059 T^{2} - \)\(25\!\cdots\!14\)\( T^{3} - \)\(15\!\cdots\!72\)\( T^{4} - \)\(25\!\cdots\!14\)\( p^{9} T^{5} - 135717001842620059 p^{18} T^{6} + 111456714 p^{27} T^{7} + p^{36} T^{8} \)
89$D_{4}$ \( ( 1 - 1234260864 T + 1048912835047912942 T^{2} - 1234260864 p^{9} T^{3} + p^{18} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 165265682 T - 1184969523064002191 T^{2} + \)\(50\!\cdots\!58\)\( T^{3} + \)\(87\!\cdots\!28\)\( T^{4} + \)\(50\!\cdots\!58\)\( p^{9} T^{5} - 1184969523064002191 p^{18} T^{6} - 165265682 p^{27} T^{7} + p^{36} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.68390287091958456682903705806, −7.14464400195652855746325709178, −7.05614515196230012629004860596, −6.51032631181779491031998972955, −6.44813004179730690151585582550, −6.43004072801749615756716718553, −6.06300899637172066295070601395, −5.63794253517252161633679577745, −5.31470428452116450833058445695, −5.01032001198377129745758362524, −4.95199039449439957197069616583, −4.36590006330289050622347522117, −4.33856983735446481254992765688, −4.01508531570046178474103775336, −3.68753842528546031449783534150, −3.06677142682634344793477759914, −3.02012200214426468925069950359, −2.90257231899001965834224398037, −2.32223147800540658489180878838, −1.99974453416099025066444453946, −1.91453886815548385191870465935, −1.22639325301539482060048908926, −0.73898112121128512065992394696, −0.59121614238830684353448917240, −0.33441065753402513695287010090, 0.33441065753402513695287010090, 0.59121614238830684353448917240, 0.73898112121128512065992394696, 1.22639325301539482060048908926, 1.91453886815548385191870465935, 1.99974453416099025066444453946, 2.32223147800540658489180878838, 2.90257231899001965834224398037, 3.02012200214426468925069950359, 3.06677142682634344793477759914, 3.68753842528546031449783534150, 4.01508531570046178474103775336, 4.33856983735446481254992765688, 4.36590006330289050622347522117, 4.95199039449439957197069616583, 5.01032001198377129745758362524, 5.31470428452116450833058445695, 5.63794253517252161633679577745, 6.06300899637172066295070601395, 6.43004072801749615756716718553, 6.44813004179730690151585582550, 6.51032631181779491031998972955, 7.05614515196230012629004860596, 7.14464400195652855746325709178, 7.68390287091958456682903705806

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