Properties

Label 8-637e4-1.1-c1e4-0-6
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  − 6·2-s + 17·4-s − 30·8-s + 6·9-s + 12·11-s + 40·16-s − 36·18-s − 72·22-s − 8·23-s − 6·25-s − 2·29-s − 54·32-s + 102·36-s − 6·37-s − 12·43-s + 204·44-s + 48·46-s + 36·50-s + 20·53-s + 12·58-s + 79·64-s + 48·67-s − 36·71-s − 180·72-s + 36·74-s − 24·79-s + 9·81-s + ⋯
L(s)  = 1  − 4.24·2-s + 17/2·4-s − 10.6·8-s + 2·9-s + 3.61·11-s + 10·16-s − 8.48·18-s − 15.3·22-s − 1.66·23-s − 6/5·25-s − 0.371·29-s − 9.54·32-s + 17·36-s − 0.986·37-s − 1.82·43-s + 30.7·44-s + 7.07·46-s + 5.09·50-s + 2.74·53-s + 1.57·58-s + 79/8·64-s + 5.86·67-s − 4.27·71-s − 21.2·72-s + 4.18·74-s − 2.70·79-s + 81-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

\(L(1)\) \(\approx\) \(0.2059407733\)
\(L(\frac12)\) \(\approx\) \(0.2059407733\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
good2$C_2^2$ \( ( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
3$C_2$ \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
5$C_2^2$ \( ( 1 + 3 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 5 T^{2} - 264 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^3$ \( 1 - 14 T^{2} - 165 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - p T^{2} )^{4} \)
37$C_2^2$ \( ( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \)
41$C_2^3$ \( 1 + 69 T^{2} + 3080 T^{4} + 69 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2^2$ \( ( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \)
53$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
59$C_2^3$ \( 1 + 66 T^{2} + 875 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^3$ \( 1 - 83 T^{2} + 3168 T^{4} - 83 p^{2} T^{6} + p^{4} T^{8} \)
67$C_2^2$ \( ( 1 - 24 T + 259 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 18 T + 179 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \)
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{4} \)
83$C_2^2$ \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 126 T^{2} + 7955 T^{4} + 126 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^3$ \( 1 + 142 T^{2} + 10755 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \)
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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

−7.980029677477025086783119041861, −7.33662940912575958561207768540, −7.28794309194263491909331314502, −7.17662170439854535957314117327, −6.86981878470427440821554762112, −6.68379195279553670757359955369, −6.59621244688971070378589983007, −6.21101416090532414378092411036, −5.81097095487179657548977896476, −5.70310504524885945680930765128, −5.47332365673608099628079144604, −4.89692257069560783077233991617, −4.54563447999940184406540278293, −4.13504431006134506523862720970, −4.00872987438536625854239342655, −3.80574935754804569119346087586, −3.74091289655486929302239605029, −3.19363821162491801652653090674, −2.65451630707336712811170433743, −1.94727753319219183722341688458, −1.75789193504972211826357711751, −1.65438281468056873717286811131, −1.19814982068977736533655416662, −1.02563077160050588388763888264, −0.36174492751192283920582065737, 0.36174492751192283920582065737, 1.02563077160050588388763888264, 1.19814982068977736533655416662, 1.65438281468056873717286811131, 1.75789193504972211826357711751, 1.94727753319219183722341688458, 2.65451630707336712811170433743, 3.19363821162491801652653090674, 3.74091289655486929302239605029, 3.80574935754804569119346087586, 4.00872987438536625854239342655, 4.13504431006134506523862720970, 4.54563447999940184406540278293, 4.89692257069560783077233991617, 5.47332365673608099628079144604, 5.70310504524885945680930765128, 5.81097095487179657548977896476, 6.21101416090532414378092411036, 6.59621244688971070378589983007, 6.68379195279553670757359955369, 6.86981878470427440821554762112, 7.17662170439854535957314117327, 7.28794309194263491909331314502, 7.33662940912575958561207768540, 7.980029677477025086783119041861

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