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Properties

Label 8-10e8-1.1-c13e4-0-0
Degree $8$
Conductor $100000000$
Sign $1$
Analytic cond. $1.32214\times 10^{8}$
Root an. cond. $10.3552$
Motivic weight $13$
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  + 1.22e6·9-s + 1.70e7·11-s − 4.77e8·19-s + 1.73e10·29-s − 8.06e9·31-s + 4.33e10·41-s + 4.66e10·49-s − 3.66e11·59-s + 1.37e11·61-s − 5.99e11·71-s + 5.13e11·79-s − 2.48e12·81-s − 1.16e13·89-s + 2.07e13·99-s + 8.97e12·101-s + 6.02e13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.10e14·169-s − 5.84e14·171-s + ⋯
L(s)  = 1  + 0.766·9-s + 2.89·11-s − 2.33·19-s + 5.40·29-s − 1.63·31-s + 1.42·41-s + 0.481·49-s − 1.13·59-s + 0.342·61-s − 0.555·71-s + 0.237·79-s − 0.978·81-s − 2.49·89-s + 2.21·99-s + 0.841·101-s + 1.74·121-s + 2.34·169-s − 1.78·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 5^{8}\)
Sign: $1$
Analytic conductor: \(1.32214\times 10^{8}\)
Root analytic conductor: \(10.3552\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 5^{8} ,\ ( \ : 13/2, 13/2, 13/2, 13/2 ),\ 1 )\)

Particular Values

\(L(7)\) \(\approx\) \(3.106257054\)
\(L(\frac12)\) \(\approx\) \(3.106257054\)
\(L(\frac{15}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$D_4\times C_2$ \( 1 - 135836 p^{2} T^{2} + 606893782 p^{8} T^{4} - 135836 p^{28} T^{6} + p^{52} T^{8} \)
7$D_4\times C_2$ \( 1 - 46617600140 T^{2} - \)\(10\!\cdots\!98\)\( p^{2} T^{4} - 46617600140 p^{26} T^{6} + p^{52} T^{8} \)
11$D_{4}$ \( ( 1 - 8500080 T + 7114598702642 p T^{2} - 8500080 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 710832910911404 T^{2} + \)\(25\!\cdots\!22\)\( T^{4} - 710832910911404 p^{26} T^{6} + p^{52} T^{8} \)
17$D_4\times C_2$ \( 1 - 19100363106124796 T^{2} + \)\(18\!\cdots\!42\)\( T^{4} - 19100363106124796 p^{26} T^{6} + p^{52} T^{8} \)
19$D_{4}$ \( ( 1 + 238996312 T + 65310799772488854 T^{2} + 238996312 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 23953020456264140 T^{2} - \)\(48\!\cdots\!22\)\( T^{4} - 23953020456264140 p^{26} T^{6} + p^{52} T^{8} \)
29$D_{4}$ \( ( 1 - 8663268228 T + 35441505443412290974 T^{2} - 8663268228 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 4033525448 T + 49011907812955175358 T^{2} + 4033525448 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 63924311811543252620 T^{2} + \)\(24\!\cdots\!18\)\( T^{4} - 63924311811543252620 p^{26} T^{6} + p^{52} T^{8} \)
41$D_{4}$ \( ( 1 - 21661691172 T + \)\(38\!\cdots\!38\)\( T^{2} - 21661691172 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - \)\(36\!\cdots\!72\)\( T^{2} + \)\(84\!\cdots\!94\)\( T^{4} - \)\(36\!\cdots\!72\)\( p^{26} T^{6} + p^{52} T^{8} \)
47$D_4\times C_2$ \( 1 - \)\(97\!\cdots\!56\)\( T^{2} + \)\(82\!\cdots\!42\)\( T^{4} - \)\(97\!\cdots\!56\)\( p^{26} T^{6} + p^{52} T^{8} \)
53$D_4\times C_2$ \( 1 - \)\(88\!\cdots\!00\)\( T^{2} + \)\(33\!\cdots\!58\)\( T^{4} - \)\(88\!\cdots\!00\)\( p^{26} T^{6} + p^{52} T^{8} \)
59$D_{4}$ \( ( 1 + 183106613304 T + \)\(20\!\cdots\!62\)\( T^{2} + 183106613304 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 68904440284 T + \)\(40\!\cdots\!26\)\( T^{2} - 68904440284 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - \)\(64\!\cdots\!40\)\( T^{2} + \)\(49\!\cdots\!38\)\( T^{4} - \)\(64\!\cdots\!40\)\( p^{26} T^{6} + p^{52} T^{8} \)
71$D_{4}$ \( ( 1 + 299938755864 T - \)\(98\!\cdots\!54\)\( T^{2} + 299938755864 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - \)\(41\!\cdots\!04\)\( T^{2} + \)\(84\!\cdots\!82\)\( T^{4} - \)\(41\!\cdots\!04\)\( p^{26} T^{6} + p^{52} T^{8} \)
79$D_{4}$ \( ( 1 - 256669983344 T + \)\(90\!\cdots\!62\)\( T^{2} - 256669983344 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - \)\(20\!\cdots\!60\)\( T^{2} + \)\(21\!\cdots\!38\)\( T^{4} - \)\(20\!\cdots\!60\)\( p^{26} T^{6} + p^{52} T^{8} \)
89$D_{4}$ \( ( 1 + 5848281889428 T + \)\(40\!\cdots\!34\)\( T^{2} + 5848281889428 p^{13} T^{3} + p^{26} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - \)\(18\!\cdots\!20\)\( T^{2} + \)\(17\!\cdots\!58\)\( T^{4} - \)\(18\!\cdots\!20\)\( p^{26} T^{6} + p^{52} T^{8} \)
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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

−7.79878620463529558848862218008, −7.07967040779076752884979236851, −7.07847200435750658581674858739, −6.68797623072685377633494262875, −6.47222844671450315263660082033, −6.46534894086221330621886639486, −6.06828521326698128132148391492, −5.69261058303066267773983034561, −5.44109900447131676242325935434, −4.62563831076631296910510977714, −4.56167118472422686249834727675, −4.45298172255010272093527924654, −4.27651681861515713020234930617, −3.75612698987786262723464074239, −3.68050445430675430281071690630, −3.06792202705217478780022076672, −2.83148752694557403538025877068, −2.53911149475986970468506777047, −2.02911997002883593074434333288, −1.82400773916864627949068876949, −1.44276177910425409429317798237, −1.12355961539355495980459416030, −0.918780003363450125614230089576, −0.72364811462909773596279575023, −0.15063618339844572858379858047, 0.15063618339844572858379858047, 0.72364811462909773596279575023, 0.918780003363450125614230089576, 1.12355961539355495980459416030, 1.44276177910425409429317798237, 1.82400773916864627949068876949, 2.02911997002883593074434333288, 2.53911149475986970468506777047, 2.83148752694557403538025877068, 3.06792202705217478780022076672, 3.68050445430675430281071690630, 3.75612698987786262723464074239, 4.27651681861515713020234930617, 4.45298172255010272093527924654, 4.56167118472422686249834727675, 4.62563831076631296910510977714, 5.44109900447131676242325935434, 5.69261058303066267773983034561, 6.06828521326698128132148391492, 6.46534894086221330621886639486, 6.47222844671450315263660082033, 6.68797623072685377633494262875, 7.07847200435750658581674858739, 7.07967040779076752884979236851, 7.79878620463529558848862218008

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