Properties

Label 8-2352e4-1.1-c2e4-0-10
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  + 12·5-s − 6·9-s − 36·17-s + 64·25-s + 48·29-s + 64·37-s + 132·41-s − 72·45-s + 192·53-s + 264·61-s − 24·73-s + 27·81-s − 432·85-s + 276·89-s + 600·97-s − 252·101-s − 280·109-s − 168·113-s + 256·121-s + 372·125-s + 127-s + 131-s + 137-s + 139-s + 576·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 12/5·5-s − 2/3·9-s − 2.11·17-s + 2.55·25-s + 1.65·29-s + 1.72·37-s + 3.21·41-s − 8/5·45-s + 3.62·53-s + 4.32·61-s − 0.328·73-s + 1/3·81-s − 5.08·85-s + 3.10·89-s + 6.18·97-s − 2.49·101-s − 2.56·109-s − 1.48·113-s + 2.11·121-s + 2.97·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.97·145-s + 0.00671·149-s + 0.00662·151-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

\(L(\frac{3}{2})\) \(\approx\) \(18.52378418\)
\(L(\frac12)\) \(\approx\) \(18.52378418\)
\(L(2)\) 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$D_{4}$ \( ( 1 - 6 T + 22 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 256 T^{2} + 44334 T^{4} - 256 p^{4} T^{6} + p^{8} T^{8} \)
13$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
17$D_{4}$ \( ( 1 + 18 T + 622 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
19$D_4\times C_2$ \( 1 - 28 p T^{2} + 310086 T^{4} - 28 p^{5} T^{6} + p^{8} T^{8} \)
23$D_4\times C_2$ \( 1 - 1168 T^{2} + 739566 T^{4} - 1168 p^{4} T^{6} + p^{8} T^{8} \)
29$D_{4}$ \( ( 1 - 24 T + 494 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 - 76 p T^{2} + 2701926 T^{4} - 76 p^{5} T^{6} + p^{8} T^{8} \)
37$D_{4}$ \( ( 1 - 32 T + 1662 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
41$D_{4}$ \( ( 1 - 66 T + 4414 T^{2} - 66 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
43$C_2^2$ \( ( 1 - 2930 T^{2} + p^{4} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 4420 T^{2} + 11574534 T^{4} - 4420 p^{4} T^{6} + p^{8} T^{8} \)
53$D_{4}$ \( ( 1 - 96 T + 6590 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 5186 T^{2} + p^{4} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 - 132 T + 10466 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 9748 T^{2} + 62331846 T^{4} - 9748 p^{4} T^{6} + p^{8} T^{8} \)
71$D_4\times C_2$ \( 1 - 14320 T^{2} + 100457262 T^{4} - 14320 p^{4} T^{6} + p^{8} T^{8} \)
73$D_{4}$ \( ( 1 + 12 T + 9362 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
79$D_4\times C_2$ \( 1 - 1972 T^{2} - 41007642 T^{4} - 1972 p^{4} T^{6} + p^{8} T^{8} \)
83$D_4\times C_2$ \( 1 - 20548 T^{2} + 188196006 T^{4} - 20548 p^{4} T^{6} + p^{8} T^{8} \)
89$D_{4}$ \( ( 1 - 138 T + 14350 T^{2} - 138 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
97$D_{4}$ \( ( 1 - 300 T + 39986 T^{2} - 300 p^{2} T^{3} + p^{4} T^{4} )^{2} \)
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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

−6.31976577567472739503370425911, −5.83628324918865413870093828695, −5.74376733350372437952058023843, −5.66616781807950182003719074407, −5.51543927383642575911811987539, −5.27215587215522862712307775107, −4.87603016350702272286427281518, −4.76465153890887823826987850357, −4.62032153509005319565462432694, −4.29782674744180591255982807253, −4.01196415197555433480959668295, −3.93725313556305781825995639281, −3.67754658176515912515547159333, −3.33760702428357902226630605353, −2.73042008745469381300123129485, −2.70222769996513676642567885309, −2.67876800519266624528750729045, −2.31215675535558718170582129274, −2.14448872057675879423244490840, −1.92730300080200392340613779413, −1.81098434828358126156703514361, −0.969271288031785980815988586064, −0.959590973427468262272272259386, −0.60859400626210769332674704209, −0.55929956857743725713379986441, 0.55929956857743725713379986441, 0.60859400626210769332674704209, 0.959590973427468262272272259386, 0.969271288031785980815988586064, 1.81098434828358126156703514361, 1.92730300080200392340613779413, 2.14448872057675879423244490840, 2.31215675535558718170582129274, 2.67876800519266624528750729045, 2.70222769996513676642567885309, 2.73042008745469381300123129485, 3.33760702428357902226630605353, 3.67754658176515912515547159333, 3.93725313556305781825995639281, 4.01196415197555433480959668295, 4.29782674744180591255982807253, 4.62032153509005319565462432694, 4.76465153890887823826987850357, 4.87603016350702272286427281518, 5.27215587215522862712307775107, 5.51543927383642575911811987539, 5.66616781807950182003719074407, 5.74376733350372437952058023843, 5.83628324918865413870093828695, 6.31976577567472739503370425911

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