Properties

Label 8-637e4-1.1-c1e4-0-3
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 − 12·9-s + 40·16-s − 72·18-s − 8·23-s + 3·25-s − 2·29-s + 54·32-s − 204·36-s + 6·37-s − 12·43-s − 48·46-s + 18·50-s − 10·53-s − 12·58-s + 79·64-s − 36·71-s − 360·72-s + 36·74-s + 12·79-s + 90·81-s − 72·86-s − 136·92-s + 51·100-s − 60·106-s + ⋯
L(s)  = 1  + 4.24·2-s + 17/2·4-s + 10.6·8-s − 4·9-s + 10·16-s − 16.9·18-s − 1.66·23-s + 3/5·25-s − 0.371·29-s + 9.54·32-s − 34·36-s + 0.986·37-s − 1.82·43-s − 7.07·46-s + 2.54·50-s − 1.37·53-s − 1.57·58-s + 79/8·64-s − 4.27·71-s − 42.4·72-s + 4.18·74-s + 1.35·79-s + 10·81-s − 7.76·86-s − 14.1·92-s + 5.09·100-s − 5.82·106-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

\(L(1)\) \(\approx\) \(5.148519332\)
\(L(\frac12)\) \(\approx\) \(5.148519332\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{4} \)
5$C_2^3$ \( 1 - 3 T^{2} - 16 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 5 T^{2} - 264 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \)
19$C_2^2$ \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \)
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$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
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$C_2^3$ \( 1 + 42 T^{2} - 445 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 66 T^{2} + 875 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 + 83 T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 18 T + 179 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \)
73$C_2^3$ \( 1 + 29 T^{2} - 4488 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
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

−7.69105833834163066147662545448, −7.33063814211889802054274453764, −6.63191143533260663629911302754, −6.54672213608285406676971517527, −6.47042354969362534499273867788, −6.20517730619048813773823513123, −5.87821654605124376207972268273, −5.84518494028935359643113665102, −5.50841329425091099436613967523, −5.28934887138452257283643879287, −5.23068533018641320486376213984, −5.15606683487691802947533179462, −4.50105566736686953522854337611, −4.42400311598368395792882317043, −4.39748896659390997246022065177, −3.76588395505877463212563529608, −3.60263426001996451256452542916, −3.51414917348911455170482919148, −3.01153593017361447647960157986, −2.96665323100623014057856891250, −2.63937966139072992820350387418, −2.55833679511662970504883023048, −1.87121257033801735848186767441, −1.45624701312338406379266520779, −0.27538367513006075059683810003, 0.27538367513006075059683810003, 1.45624701312338406379266520779, 1.87121257033801735848186767441, 2.55833679511662970504883023048, 2.63937966139072992820350387418, 2.96665323100623014057856891250, 3.01153593017361447647960157986, 3.51414917348911455170482919148, 3.60263426001996451256452542916, 3.76588395505877463212563529608, 4.39748896659390997246022065177, 4.42400311598368395792882317043, 4.50105566736686953522854337611, 5.15606683487691802947533179462, 5.23068533018641320486376213984, 5.28934887138452257283643879287, 5.50841329425091099436613967523, 5.84518494028935359643113665102, 5.87821654605124376207972268273, 6.20517730619048813773823513123, 6.47042354969362534499273867788, 6.54672213608285406676971517527, 6.63191143533260663629911302754, 7.33063814211889802054274453764, 7.69105833834163066147662545448

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