Properties

Label 8-2352e4-1.1-c2e4-0-3
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  − 2.39·5-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

\(L(\frac{3}{2})\) \(\approx\) \(0.3515015605\)
\(L(\frac12)\) \(\approx\) \(0.3515015605\)
\(L(2)\) 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

−6.04319597279205545320888013024, −5.95123021104990982877913028511, −5.75781134747297443079710365070, −5.61003578996277685345063551069, −5.34643263772343069831866401664, −5.13040969104617934312059767643, −4.94705670332891876028479293632, −4.54776946015006314750175925608, −4.41650007776904485662463424224, −4.24688699426847388797794547727, −4.24387553415347940668954524229, −3.58790847231639284981334313879, −3.54741446404863325809019468598, −3.49522375895066060177525853860, −3.33517752338844294779687500726, −2.81171579167240184761658707562, −2.67872739408659981749653836686, −2.56303265906894222398659952307, −2.33759439015189382288009389951, −1.43330943212636808649736477247, −1.42919150021129011898515935730, −1.29579961334372077832155326038, −0.897730382163920406899777968960, −0.36890880631222701337140667921, −0.12713160857264115097593821323, 0.12713160857264115097593821323, 0.36890880631222701337140667921, 0.897730382163920406899777968960, 1.29579961334372077832155326038, 1.42919150021129011898515935730, 1.43330943212636808649736477247, 2.33759439015189382288009389951, 2.56303265906894222398659952307, 2.67872739408659981749653836686, 2.81171579167240184761658707562, 3.33517752338844294779687500726, 3.49522375895066060177525853860, 3.54741446404863325809019468598, 3.58790847231639284981334313879, 4.24387553415347940668954524229, 4.24688699426847388797794547727, 4.41650007776904485662463424224, 4.54776946015006314750175925608, 4.94705670332891876028479293632, 5.13040969104617934312059767643, 5.34643263772343069831866401664, 5.61003578996277685345063551069, 5.75781134747297443079710365070, 5.95123021104990982877913028511, 6.04319597279205545320888013024

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