Properties

Label 8-637e4-1.1-c1e4-0-15
Degree $8$
Conductor $164648481361$
Sign $1$
Analytic cond. $669.369$
Root an. cond. $2.25532$
Motivic weight $1$
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  − 3·2-s + 6·3-s + 5·4-s + 3·5-s − 18·6-s − 6·8-s + 13·9-s − 9·10-s − 6·11-s + 30·12-s + 10·13-s + 18·15-s + 4·16-s + 6·17-s − 39·18-s − 6·19-s + 15·20-s + 18·22-s − 36·24-s + 11·25-s − 30·26-s − 3·29-s − 54·30-s + 4·31-s − 36·33-s − 18·34-s + 65·36-s + ⋯
L(s)  = 1  − 2.12·2-s + 3.46·3-s + 5/2·4-s + 1.34·5-s − 7.34·6-s − 2.12·8-s + 13/3·9-s − 2.84·10-s − 1.80·11-s + 8.66·12-s + 2.77·13-s + 4.64·15-s + 16-s + 1.45·17-s − 9.19·18-s − 1.37·19-s + 3.35·20-s + 3.83·22-s − 7.34·24-s + 11/5·25-s − 5.88·26-s − 0.557·29-s − 9.85·30-s + 0.718·31-s − 6.26·33-s − 3.08·34-s + 65/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(7^{8} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(669.369\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 7^{8} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.623271731\)
\(L(\frac12)\) \(\approx\) \(3.623271731\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
good2$C_2^2$$\times$$C_2^2$ \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \)
3$D_{4}$ \( ( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2} \)
5$D_4\times C_2$ \( 1 - 3 T - 2 T^{2} - 3 T^{3} + 51 T^{4} - 3 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
11$D_{4}$ \( ( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 6 T + 13 T^{2} + 66 T^{3} - 372 T^{4} + 66 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2^3$ \( 1 - 26 T^{2} + 147 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \)
29$D_4\times C_2$ \( 1 + 3 T - 20 T^{2} - 3 p T^{3} - 9 p T^{4} - 3 p^{2} T^{5} - 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
31$D_4\times C_2$ \( 1 - 4 T - 5 T^{2} + 164 T^{3} - 1016 T^{4} + 164 p T^{5} - 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 6 T - 50 T^{2} - 24 T^{3} + 4239 T^{4} - 24 p T^{5} - 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
43$D_4\times C_2$ \( 1 + 5 T + 34 T^{2} - 475 T^{3} - 2843 T^{4} - 475 p T^{5} + 34 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2^3$ \( 1 - 89 T^{2} + 5712 T^{4} - 89 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 12 T + 7 T^{2} + 372 T^{3} + 8328 T^{4} + 372 p T^{5} + 7 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2^3$ \( 1 - 113 T^{2} + 9288 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
67$D_{4}$ \( ( 1 + 12 T + 125 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_4\times C_2$ \( 1 - 6 T + 10 T^{2} + 696 T^{3} - 6921 T^{4} + 696 p T^{5} + 10 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2$ \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \)
83$C_2^2$ \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 - 21 T + 184 T^{2} - 1659 T^{3} + 18879 T^{4} - 1659 p T^{5} + 184 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
97$D_4\times C_2$ \( 1 - 31 T + 538 T^{2} - 7099 T^{3} + 76303 T^{4} - 7099 p T^{5} + 538 p^{2} T^{6} - 31 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.77237389412935278309098032333, −7.64085279074008313651878142962, −7.42160219920500983959718446407, −7.33245524479565063382594329955, −6.70509628044687772080075216176, −6.56541196484511406684760788314, −6.34658590039899400257157314856, −6.06284926775233439740003444822, −5.79443547781879640904547055231, −5.58866769036970696979478271569, −5.36093236102240234558330182428, −5.00768651332673871415454772710, −4.55865679378103979402191534661, −4.21101525129403557638109840884, −3.89385194790681835912585836718, −3.29760388512439541783887985672, −3.15798453148614947837328791181, −3.13926268433337969265276379484, −3.11789242988367805866590046499, −2.39745058276287860794435008122, −2.17566062096431827490513336240, −2.04132488132278653363383979846, −1.72636254295764892633252070540, −1.23235897296191225582736320914, −0.59013149066770062539677255264, 0.59013149066770062539677255264, 1.23235897296191225582736320914, 1.72636254295764892633252070540, 2.04132488132278653363383979846, 2.17566062096431827490513336240, 2.39745058276287860794435008122, 3.11789242988367805866590046499, 3.13926268433337969265276379484, 3.15798453148614947837328791181, 3.29760388512439541783887985672, 3.89385194790681835912585836718, 4.21101525129403557638109840884, 4.55865679378103979402191534661, 5.00768651332673871415454772710, 5.36093236102240234558330182428, 5.58866769036970696979478271569, 5.79443547781879640904547055231, 6.06284926775233439740003444822, 6.34658590039899400257157314856, 6.56541196484511406684760788314, 6.70509628044687772080075216176, 7.33245524479565063382594329955, 7.42160219920500983959718446407, 7.64085279074008313651878142962, 7.77237389412935278309098032333

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