Properties

Label 8-2352e4-1.1-c2e4-0-5
Degree $8$
Conductor $3.060\times 10^{13}$
Sign $1$
Analytic cond. $1.68690\times 10^{7}$
Root an. cond. $8.00545$
Motivic weight $2$
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  − 6·9-s − 24·11-s + 48·23-s + 28·25-s − 44·37-s − 28·43-s − 240·53-s + 220·67-s − 312·71-s − 20·79-s + 27·81-s + 144·99-s − 576·107-s + 580·109-s + 168·113-s − 124·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 98·169-s + 173-s + ⋯
L(s)  = 1  − 2/3·9-s − 2.18·11-s + 2.08·23-s + 1.11·25-s − 1.18·37-s − 0.651·43-s − 4.52·53-s + 3.28·67-s − 4.39·71-s − 0.253·79-s + 1/3·81-s + 1.45·99-s − 5.38·107-s + 5.32·109-s + 1.48·113-s − 1.02·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.579·169-s + 0.00578·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.68690\times 10^{7}\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2352} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5867941603\)
\(L(\frac12)\) \(\approx\) \(0.5867941603\)
\(L(2)\) 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 \( 1 \)
3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 - 28 T^{2} + 294 T^{4} - 28 p^{4} T^{6} + p^{8} T^{8} \)
11$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{4} \)
13$D_4\times C_2$ \( 1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} \)
17$D_4\times C_2$ \( 1 - 724 T^{2} + 279654 T^{4} - 724 p^{4} T^{6} + p^{8} T^{8} \)
19$D_4\times C_2$ \( 1 - 58 p T^{2} + 550131 T^{4} - 58 p^{5} T^{6} + p^{8} T^{8} \)
23$D_{4}$ \( ( 1 - 24 T + 554 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 1678 T^{2} + 1875 p^{2} T^{4} - 1678 p^{4} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 + 22 T + 2571 T^{2} + 22 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8} \)
43$D_{4}$ \( ( 1 + 14 T + 3675 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 5884 T^{2} + 18275334 T^{4} - 5884 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 + 120 T + 8570 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$D_4\times C_2$ \( 1 - 11476 T^{2} + 56933574 T^{4} - 11476 p^{4} T^{6} + p^{8} T^{8} \)
61$D_4\times C_2$ \( 1 - 13252 T^{2} + 70932006 T^{4} - 13252 p^{4} T^{6} + p^{8} T^{8} \)
67$D_{4}$ \( ( 1 - 110 T + 8475 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
71$D_{4}$ \( ( 1 + 156 T + 178 p T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 1390 T^{2} + 43340307 T^{4} - 1390 p^{4} T^{6} + p^{8} T^{8} \)
79$D_{4}$ \( ( 1 + 10 T + 3795 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8} \)
89$C_2^2$ \( ( 1 - 15410 T^{2} + p^{4} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8} \)
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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

−6.10780650640862366163714982280, −5.95273213458064814238941412921, −5.80047357716409569236947456336, −5.46558260536548656582654618917, −5.26798447902563472222446492105, −5.12921909690492362920680454393, −5.09110165261112585239955825547, −4.70743695271710049972874439783, −4.60636002865501278824636055540, −4.35911050047959865427445056936, −4.25790476428004183197531717691, −3.64933270463664985134029901241, −3.42183568417264908393510987487, −3.22321917332194049799690789813, −3.15450290194795871087312979783, −2.94731347048548606466473220012, −2.71469195851598807289828331625, −2.33944387392814605599131777472, −2.24849693810977941160945183763, −1.82013981704763596400555201247, −1.44553868120603611740888883098, −1.27101957041702755783803140690, −0.936650219279052917567870804100, −0.36738784913036126912468234938, −0.14452239273882443771199750249, 0.14452239273882443771199750249, 0.36738784913036126912468234938, 0.936650219279052917567870804100, 1.27101957041702755783803140690, 1.44553868120603611740888883098, 1.82013981704763596400555201247, 2.24849693810977941160945183763, 2.33944387392814605599131777472, 2.71469195851598807289828331625, 2.94731347048548606466473220012, 3.15450290194795871087312979783, 3.22321917332194049799690789813, 3.42183568417264908393510987487, 3.64933270463664985134029901241, 4.25790476428004183197531717691, 4.35911050047959865427445056936, 4.60636002865501278824636055540, 4.70743695271710049972874439783, 5.09110165261112585239955825547, 5.12921909690492362920680454393, 5.26798447902563472222446492105, 5.46558260536548656582654618917, 5.80047357716409569236947456336, 5.95273213458064814238941412921, 6.10780650640862366163714982280

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