Properties

Label 8-690e4-1.1-c1e4-0-0
Degree $8$
Conductor $226671210000$
Sign $1$
Analytic cond. $921.520$
Root an. cond. $2.34727$
Motivic weight $1$
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  − 2·4-s − 2·9-s + 3·16-s − 8·19-s − 8·25-s + 8·29-s + 24·31-s + 4·36-s − 16·41-s + 20·49-s − 8·61-s − 4·64-s + 16·76-s − 8·79-s + 3·81-s − 48·89-s + 16·100-s + 8·101-s − 56·109-s − 16·116-s − 8·121-s − 48·124-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + ⋯
L(s)  = 1  − 4-s − 2/3·9-s + 3/4·16-s − 1.83·19-s − 8/5·25-s + 1.48·29-s + 4.31·31-s + 2/3·36-s − 2.49·41-s + 20/7·49-s − 1.02·61-s − 1/2·64-s + 1.83·76-s − 0.900·79-s + 1/3·81-s − 5.08·89-s + 8/5·100-s + 0.796·101-s − 5.36·109-s − 1.48·116-s − 0.727·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(921.520\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{690} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.2311064812\)
\(L(\frac12)\) \(\approx\) \(0.2311064812\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( ( 1 + T^{2} )^{2} \)
3$C_2$ \( ( 1 + T^{2} )^{2} \)
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
good7$C_2^2$ \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \)
17$D_4\times C_2$ \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
29$D_{4}$ \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{4} \)
37$D_4\times C_2$ \( 1 - 136 T^{2} + 7330 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 64 T^{2} + 2130 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \)
47$D_4\times C_2$ \( 1 - 164 T^{2} + 11014 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \)
53$D_4\times C_2$ \( 1 + 80 T^{2} + 6066 T^{4} + 80 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2^2$ \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 4 T + 124 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 256 T^{2} + 25330 T^{4} - 256 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2^2$ \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 220 T^{2} + 22246 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \)
79$D_{4}$ \( ( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 164 T^{2} + p^{2} T^{4} )^{2} \)
89$D_{4}$ \( ( 1 + 24 T + 314 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.59037229561093858775508131553, −7.54742181514054021058892365220, −6.83621072564704550469791702038, −6.71683285618193204711957169424, −6.58169695763093165944208686867, −6.39984659361714939097998460042, −6.29298581092539364957621248367, −5.71823973354859464208238041088, −5.65294831464020631034018359279, −5.35001672789286272103546967679, −5.24179012591949861526600923519, −4.67086208995320879786151109404, −4.65514637526271805955202461886, −4.33194506087654234908927267274, −4.04468690662959987181201492212, −3.95168773788431369212375910229, −3.73045055904861672129954303485, −2.94488202302456260376400984208, −2.91712678565351579783689073692, −2.58579478179252669497453466095, −2.51405555827995043101920717066, −1.82728893650453217587657874478, −1.28810364690742197723188578930, −1.09905093290572192911390018785, −0.14945565433026806110664592980, 0.14945565433026806110664592980, 1.09905093290572192911390018785, 1.28810364690742197723188578930, 1.82728893650453217587657874478, 2.51405555827995043101920717066, 2.58579478179252669497453466095, 2.91712678565351579783689073692, 2.94488202302456260376400984208, 3.73045055904861672129954303485, 3.95168773788431369212375910229, 4.04468690662959987181201492212, 4.33194506087654234908927267274, 4.65514637526271805955202461886, 4.67086208995320879786151109404, 5.24179012591949861526600923519, 5.35001672789286272103546967679, 5.65294831464020631034018359279, 5.71823973354859464208238041088, 6.29298581092539364957621248367, 6.39984659361714939097998460042, 6.58169695763093165944208686867, 6.71683285618193204711957169424, 6.83621072564704550469791702038, 7.54742181514054021058892365220, 7.59037229561093858775508131553

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