Properties

Degree $8$
Conductor $15752961$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 47·16-s − 500·25-s + 686·49-s + 2.96e3·67-s − 5.53e3·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯
L(s)  = 1  + 0.734·16-s − 4·25-s + 2·49-s + 5.39·67-s − 7.88·79-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + 0.000288·229-s + 0.000281·233-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15752961 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(15752961\)    =    \(3^{8} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{63} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 15752961,\ (\ :3/2, 3/2, 3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.64626\)
\(L(\frac12)\) \(\approx\) \(1.64626\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
7$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
good2$C_2^3$ \( 1 - 47 T^{4} + p^{12} T^{8} \)
5$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
11$C_2^3$ \( 1 + 306322 T^{4} + p^{12} T^{8} \)
13$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
17$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
19$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
23$C_2^3$ \( 1 + 220762978 T^{4} + p^{12} T^{8} \)
29$C_2^3$ \( 1 - 739273358 T^{4} + p^{12} T^{8} \)
31$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 101194 T^{2} + p^{6} T^{4} )^{2} \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 126614 T^{2} + p^{6} T^{4} )^{2} \)
47$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
53$C_2^3$ \( 1 - 41794002542 T^{4} + p^{12} T^{8} \)
59$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
61$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
67$C_2$ \( ( 1 - 740 T + p^{3} T^{2} )^{4} \)
71$C_2^3$ \( 1 - 197404987358 T^{4} + p^{12} T^{8} \)
73$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
79$C_2$ \( ( 1 + 1384 T + p^{3} T^{2} )^{4} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
89$C_2$ \( ( 1 + p^{3} T^{2} )^{4} \)
97$C_2$ \( ( 1 - p^{3} T^{2} )^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.51009801635924384486448848235, −10.13092149744954092187930118706, −9.944500625117449408980862581083, −9.923477192646726845296226898316, −9.420753723152216723632914325286, −9.241450780954893282778454900706, −8.402392327330469460790107586973, −8.389119888755360465783414245895, −8.380217848138606239718806710845, −7.65469449645162492878361295795, −7.45232098653198950654503520588, −7.08244716845334557569672896844, −6.85882528085845714350427961432, −6.13028116741244000476307604241, −5.85680225559852868161634825107, −5.59328794845104549216704373775, −5.45857681032163956187481676432, −4.63213618137427135380497545344, −4.14452910051773280477311402234, −3.93566870972386500702500289583, −3.47829341083152219219793015352, −2.76272598405608119082452877240, −2.14342547010408960919208194569, −1.61031405005102637955121769224, −0.50827161462858869577201668080, 0.50827161462858869577201668080, 1.61031405005102637955121769224, 2.14342547010408960919208194569, 2.76272598405608119082452877240, 3.47829341083152219219793015352, 3.93566870972386500702500289583, 4.14452910051773280477311402234, 4.63213618137427135380497545344, 5.45857681032163956187481676432, 5.59328794845104549216704373775, 5.85680225559852868161634825107, 6.13028116741244000476307604241, 6.85882528085845714350427961432, 7.08244716845334557569672896844, 7.45232098653198950654503520588, 7.65469449645162492878361295795, 8.380217848138606239718806710845, 8.389119888755360465783414245895, 8.402392327330469460790107586973, 9.241450780954893282778454900706, 9.420753723152216723632914325286, 9.923477192646726845296226898316, 9.944500625117449408980862581083, 10.13092149744954092187930118706, 10.51009801635924384486448848235

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