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Properties

Label 8-2624e4-1.1-c0e4-0-1
Degree $8$
Conductor $4.741\times 10^{13}$
Sign $1$
Analytic cond. $2.94092$
Root an. cond. $1.14435$
Motivic weight $0$
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  − 4·41-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-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 + 239-s + 241-s + 251-s + 257-s + ⋯
L(s)  = 1  − 4·41-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-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 + 239-s + 241-s + 251-s + 257-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 41^{4}\)
Sign: $1$
Analytic conductor: \(2.94092\)
Root analytic conductor: \(1.14435\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 41^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8829719961\)
\(L(\frac12)\) \(\approx\) \(0.8829719961\)
\(L(1)\) 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 \( 1 \)
41$C_1$ \( ( 1 + T )^{4} \)
good3$C_4\times C_2$ \( 1 + T^{8} \)
5$C_2^2$ \( ( 1 + T^{4} )^{2} \)
7$C_4\times C_2$ \( 1 + T^{8} \)
11$C_4\times C_2$ \( 1 + T^{8} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
19$C_4\times C_2$ \( 1 + T^{8} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
37$C_2^2$ \( ( 1 + T^{4} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
47$C_4\times C_2$ \( 1 + T^{8} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{4} \)
67$C_4\times C_2$ \( 1 + T^{8} \)
71$C_4\times C_2$ \( 1 + T^{8} \)
73$C_2^2$ \( ( 1 + T^{4} )^{2} \)
79$C_4\times C_2$ \( 1 + T^{8} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{4}( 1 + T )^{4} \)
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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

−6.61343986801005970337179025153, −6.15967571861287890282572768240, −6.01042521301619577128174695088, −5.93955290880403532173534392679, −5.86096665836022561928075147000, −5.26297605638245412241856276540, −5.18389419355178153047013591117, −5.08498083536035753835111024982, −4.87329265134818911642972774808, −4.82783432936900315067708350423, −4.30221010155252967085880246593, −4.13619084843869738962352474581, −4.03899756836353085265555069365, −3.63946657573238044655616759175, −3.53410783562962527827304776894, −3.18575805913176957651860779903, −3.13118067102228926691222432950, −2.85447297829314229848466532601, −2.36210336555706107002341535308, −2.34579497983169547419490833076, −1.83412772387732541258838915743, −1.79977135790871184252128714256, −1.26534155454990930009806785167, −1.20126846340212985853034214038, −0.40869646541742863523646276767, 0.40869646541742863523646276767, 1.20126846340212985853034214038, 1.26534155454990930009806785167, 1.79977135790871184252128714256, 1.83412772387732541258838915743, 2.34579497983169547419490833076, 2.36210336555706107002341535308, 2.85447297829314229848466532601, 3.13118067102228926691222432950, 3.18575805913176957651860779903, 3.53410783562962527827304776894, 3.63946657573238044655616759175, 4.03899756836353085265555069365, 4.13619084843869738962352474581, 4.30221010155252967085880246593, 4.82783432936900315067708350423, 4.87329265134818911642972774808, 5.08498083536035753835111024982, 5.18389419355178153047013591117, 5.26297605638245412241856276540, 5.86096665836022561928075147000, 5.93955290880403532173534392679, 6.01042521301619577128174695088, 6.15967571861287890282572768240, 6.61343986801005970337179025153

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