Properties

Label 8-2352e4-1.1-c2e4-0-1
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $1.68690\times 10^{7}$
Root an. cond. $8.00545$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·5-s − 6·9-s − 8·13-s − 56·17-s − 60·25-s − 56·29-s + 104·37-s + 40·41-s − 48·45-s + 40·53-s + 24·61-s − 64·65-s − 232·73-s + 27·81-s − 448·85-s − 152·89-s + 88·97-s − 152·101-s − 280·109-s − 248·113-s + 48·117-s + 164·121-s − 840·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 8/5·5-s − 2/3·9-s − 0.615·13-s − 3.29·17-s − 2.39·25-s − 1.93·29-s + 2.81·37-s + 0.975·41-s − 1.06·45-s + 0.754·53-s + 0.393·61-s − 0.984·65-s − 3.17·73-s + 1/3·81-s − 5.27·85-s − 1.70·89-s + 0.907·97-s − 1.50·101-s − 2.56·109-s − 2.19·113-s + 0.410·117-s + 1.35·121-s − 6.71·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.68690\times 10^{7}\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2341495616\)
\(L(\frac12)\) \(\approx\) \(0.2341495616\)
\(L(2)\) 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 \( 1 \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7 \( 1 \)
good5$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{4} \)
11$D_4\times C_2$ \( 1 - 164 T^{2} + 14502 T^{4} - 164 p^{4} T^{6} + p^{8} T^{8} \)
13$D_{4}$ \( ( 1 + 4 T + 6 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
17$D_{4}$ \( ( 1 + 28 T + 438 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 452 T^{2} + 225702 T^{4} - 452 p^{4} T^{6} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 1028 T^{2} + 630342 T^{4} - 1028 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 + 28 T + 534 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1154 T^{2} + p^{4} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 - 52 T + 2070 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 20 T + 3126 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 4964 T^{2} + 11621670 T^{4} - 4964 p^{4} T^{6} + p^{8} T^{8} \)
47$D_4\times C_2$ \( 1 + 764 T^{2} - 2481018 T^{4} + 764 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 - 20 T + 4374 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 12164 T^{2} + 60451302 T^{4} - 12164 p^{4} T^{6} + p^{8} T^{8} \)
61$D_{4}$ \( ( 1 - 12 T + 7142 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$C_2$ \( ( 1 - 122 T + p^{2} T^{2} )^{2}( 1 + 122 T + p^{2} T^{2} )^{2} \)
71$D_4\times C_2$ \( 1 - 8324 T^{2} + 58662342 T^{4} - 8324 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 + 116 T + 12678 T^{2} + 116 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 3074 T^{2} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 12356 T^{2} + 120698022 T^{4} - 12356 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 + 76 T + 14262 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 44 T + 13926 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−6.08259443163057832027408241112, −6.03954932343781285303028209255, −5.79101200164174183631243040242, −5.63498123265649045369784772060, −5.39635822255382059405554970722, −5.33334501240991896594506632917, −5.05662332463734613098709469025, −4.50874748815475506039213049830, −4.48428076603122448311248108415, −4.31796573525379661887647197241, −4.05595992396231138366721597447, −3.96780043886281186751728806491, −3.78250403913916373179627572036, −3.31138845408463110588275769276, −2.83910212365397189449424836219, −2.81849004310528102582249356419, −2.44972344785513905765459147608, −2.40907478784877800961567709071, −2.07904537905960221966469301763, −1.93043192497047807176378229000, −1.70422987643072422357388816771, −1.33038296407688146138724428303, −0.982614205373243722268732483066, −0.31293766490045153465538950796, −0.092324230491916657908148268011, 0.092324230491916657908148268011, 0.31293766490045153465538950796, 0.982614205373243722268732483066, 1.33038296407688146138724428303, 1.70422987643072422357388816771, 1.93043192497047807176378229000, 2.07904537905960221966469301763, 2.40907478784877800961567709071, 2.44972344785513905765459147608, 2.81849004310528102582249356419, 2.83910212365397189449424836219, 3.31138845408463110588275769276, 3.78250403913916373179627572036, 3.96780043886281186751728806491, 4.05595992396231138366721597447, 4.31796573525379661887647197241, 4.48428076603122448311248108415, 4.50874748815475506039213049830, 5.05662332463734613098709469025, 5.33334501240991896594506632917, 5.39635822255382059405554970722, 5.63498123265649045369784772060, 5.79101200164174183631243040242, 6.03954932343781285303028209255, 6.08259443163057832027408241112

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