Properties

Label 8-637e4-1.1-c1e4-0-0
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  − 4·4-s + 6·9-s + 4·16-s − 2·23-s + 3·25-s − 20·29-s − 24·36-s + 36·43-s − 22·53-s + 16·64-s − 30·79-s + 9·81-s + 8·92-s − 12·100-s − 16·107-s − 76·113-s + 80·116-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 24·144-s + 149-s + 151-s + 157-s + 163-s + ⋯
L(s)  = 1  − 2·4-s + 2·9-s + 16-s − 0.417·23-s + 3/5·25-s − 3.71·29-s − 4·36-s + 5.48·43-s − 3.02·53-s + 2·64-s − 3.37·79-s + 81-s + 0.834·92-s − 6/5·100-s − 1.54·107-s − 7.14·113-s + 7.42·116-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-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

\(L(1)\) \(\approx\) \(0.1114138752\)
\(L(\frac12)\) \(\approx\) \(0.1114138752\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
good2$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
3$C_2$ \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \)
5$C_2^3$ \( 1 - 3 T^{2} - 16 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \)
11$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
19$C_2^3$ \( 1 + 25 T^{2} + 264 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \)
23$C_2^2$ \( ( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{4} \)
31$C_2^3$ \( 1 - 55 T^{2} + 2064 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 9 T + p T^{2} )^{4} \)
47$C_2^3$ \( 1 + 81 T^{2} + 4352 T^{4} + 81 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2^2$ \( ( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \)
59$C_2^3$ \( 1 - 90 T^{2} + 4619 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \)
61$C_2^2$ \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \)
71$C_2$ \( ( 1 - p T^{2} )^{4} \)
73$C_2^3$ \( 1 + 29 T^{2} - 4488 T^{4} + 29 p^{2} T^{6} + p^{4} T^{8} \)
79$C_2^2$ \( ( 1 + 15 T + 146 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 + 159 T^{2} + p^{2} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 165 T^{2} + 19304 T^{4} + 165 p^{2} T^{6} + p^{4} T^{8} \)
97$C_2^2$ \( ( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \)
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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

−7.64896177167532344749524847071, −7.37092357734481818411126943077, −7.22736359153348885589483677072, −7.13411185950019118077099482296, −6.61661008332818910175282622763, −6.52459738983402454525999425454, −6.09822562860099551616449803030, −5.95291657749220701817875402989, −5.57885936297126169182553481463, −5.28596404846057207592207687406, −5.26791181324131184487664950479, −4.94003789492775099633991488438, −4.42546371503590112072054326494, −4.33297305190159832153240791561, −4.25846307408306885289855760786, −3.85090884608429443645664173624, −3.75388415283236017820156335754, −3.71917697280005087420116182610, −2.77698270061114537509774701645, −2.55193160801290929009739828651, −2.50081488287568496765062229290, −1.54817598131703018215100034517, −1.52139997198257147309992813825, −1.13070580468281495959922588167, −0.10949559964651829561813409596, 0.10949559964651829561813409596, 1.13070580468281495959922588167, 1.52139997198257147309992813825, 1.54817598131703018215100034517, 2.50081488287568496765062229290, 2.55193160801290929009739828651, 2.77698270061114537509774701645, 3.71917697280005087420116182610, 3.75388415283236017820156335754, 3.85090884608429443645664173624, 4.25846307408306885289855760786, 4.33297305190159832153240791561, 4.42546371503590112072054326494, 4.94003789492775099633991488438, 5.26791181324131184487664950479, 5.28596404846057207592207687406, 5.57885936297126169182553481463, 5.95291657749220701817875402989, 6.09822562860099551616449803030, 6.52459738983402454525999425454, 6.61661008332818910175282622763, 7.13411185950019118077099482296, 7.22736359153348885589483677072, 7.37092357734481818411126943077, 7.64896177167532344749524847071

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