Properties

Degree $8$
Conductor $390625$
Sign $1$
Motivic weight $8$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.85e4·11-s + 7.81e4·16-s − 4.17e6·31-s − 5.28e6·41-s − 5.75e7·61-s − 9.22e7·71-s − 7.87e7·81-s + 7.22e8·101-s + 6.14e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 3.79e9·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 3.31·11-s + 1.19·16-s − 4.51·31-s − 1.87·41-s − 4.15·61-s − 3.63·71-s − 1.82·81-s + 6.94·101-s + 2.86·121-s + 3.95·176-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(390625\)    =    \(5^{8}\)
Sign: $1$
Motivic weight: \(8\)
Character: induced by $\chi_{25} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 390625,\ (\ :4, 4, 4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.01948\)
\(L(\frac12)\) \(\approx\) \(2.01948\)
\(L(5)\) 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)$
bad5 \( 1 \)
good2$C_2^3$ \( 1 - 19543 p^{2} T^{4} + p^{32} T^{8} \)
3$C_2^3$ \( 1 + 8752462 p^{2} T^{4} + p^{32} T^{8} \)
7$C_2^3$ \( 1 - 22527334402 p^{4} T^{4} + p^{32} T^{8} \)
11$C_2$ \( ( 1 - 12132 T + p^{8} T^{2} )^{4} \)
13$C_2^3$ \( 1 + 1292678222057673218 T^{4} + p^{32} T^{8} \)
17$C_2^3$ \( 1 - 92930849881804020862 T^{4} + p^{32} T^{8} \)
19$C_2^2$ \( ( 1 - 5615301682 T^{2} + p^{16} T^{4} )^{2} \)
23$C_2^3$ \( 1 - \)\(88\!\cdots\!22\)\( T^{4} + p^{32} T^{8} \)
29$C_2^2$ \( ( 1 - 556097269022 T^{2} + p^{16} T^{4} )^{2} \)
31$C_2$ \( ( 1 + 1042808 T + p^{8} T^{2} )^{4} \)
37$C_2^3$ \( 1 - \)\(22\!\cdots\!82\)\( T^{4} + p^{32} T^{8} \)
41$C_2$ \( ( 1 + 1321128 T + p^{8} T^{2} )^{4} \)
43$C_2^3$ \( 1 - \)\(21\!\cdots\!02\)\( T^{4} + p^{32} T^{8} \)
47$C_2^3$ \( 1 - \)\(30\!\cdots\!38\)\( p^{2} T^{4} + p^{32} T^{8} \)
53$C_2^3$ \( 1 + \)\(32\!\cdots\!58\)\( T^{4} + p^{32} T^{8} \)
59$C_2^2$ \( ( 1 - 251429853077042 T^{2} + p^{16} T^{4} )^{2} \)
61$C_2$ \( ( 1 + 14393968 T + p^{8} T^{2} )^{4} \)
67$C_2^3$ \( 1 - \)\(27\!\cdots\!62\)\( T^{4} + p^{32} T^{8} \)
71$C_2$ \( ( 1 + 23065488 T + p^{8} T^{2} )^{4} \)
73$C_2^3$ \( 1 - \)\(29\!\cdots\!22\)\( T^{4} + p^{32} T^{8} \)
79$C_2^2$ \( ( 1 - 3026596265750722 T^{2} + p^{16} T^{4} )^{2} \)
83$C_2^3$ \( 1 + \)\(78\!\cdots\!38\)\( T^{4} + p^{32} T^{8} \)
89$C_2^2$ \( ( 1 - 7190374058544062 T^{2} + p^{16} T^{4} )^{2} \)
97$C_2^3$ \( 1 - \)\(11\!\cdots\!42\)\( T^{4} + p^{32} T^{8} \)
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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

−11.59787917445806493331916721336, −10.89270854844029005366075275826, −10.75923517528193821773334966735, −10.33562561084523302323539833859, −9.804950947862602499730862019986, −9.503854729516589760068196366077, −8.940562126313141866949483484668, −8.909358550712252079513390698647, −8.810606409920519879239672108156, −7.950051031464391513647461378696, −7.42269233055240161981700020501, −7.10231960563365664466626627171, −7.02420414343425529715621808214, −6.03751932657559777528337007657, −6.02492577982123154300267029597, −5.74908006895450568341471261491, −4.68859398590647797210003383307, −4.58818484252313081043069292160, −3.65382816190440702852192984635, −3.57094785846168390746734024769, −3.23481496563591788452217803052, −1.90749713065090888877954839981, −1.44747688200536942416589873753, −1.42436558255833054154923093605, −0.30000086782784413153038542258, 0.30000086782784413153038542258, 1.42436558255833054154923093605, 1.44747688200536942416589873753, 1.90749713065090888877954839981, 3.23481496563591788452217803052, 3.57094785846168390746734024769, 3.65382816190440702852192984635, 4.58818484252313081043069292160, 4.68859398590647797210003383307, 5.74908006895450568341471261491, 6.02492577982123154300267029597, 6.03751932657559777528337007657, 7.02420414343425529715621808214, 7.10231960563365664466626627171, 7.42269233055240161981700020501, 7.950051031464391513647461378696, 8.810606409920519879239672108156, 8.909358550712252079513390698647, 8.940562126313141866949483484668, 9.503854729516589760068196366077, 9.804950947862602499730862019986, 10.33562561084523302323539833859, 10.75923517528193821773334966735, 10.89270854844029005366075275826, 11.59787917445806493331916721336

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