Properties

Label 8-768e4-1.1-c2e4-0-2
Degree $8$
Conductor $347892350976$
Sign $1$
Analytic cond. $191771.$
Root an. cond. $4.57454$
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 − 72·17-s − 76·25-s + 216·41-s + 188·49-s − 40·73-s + 27·81-s + 72·89-s − 136·97-s + 360·113-s + 100·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 432·153-s + 157-s + 163-s + 167-s + 188·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2/3·9-s − 4.23·17-s − 3.03·25-s + 5.26·41-s + 3.83·49-s − 0.547·73-s + 1/3·81-s + 0.808·89-s − 1.40·97-s + 3.18·113-s + 0.826·121-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 + 2.82·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.11·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{32} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(191771.\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{32} \cdot 3^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2328151422\)
\(L(\frac12)\) \(\approx\) \(0.2328151422\)
\(L(2)\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \)
good5$C_2^2$ \( ( 1 + 38 T^{2} + p^{4} T^{4} )^{2} \)
7$C_2^2$ \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \)
11$C_2^2$ \( ( 1 - 50 T^{2} + p^{4} T^{4} )^{2} \)
13$C_2^2$ \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \)
17$C_2$ \( ( 1 + 18 T + p^{2} T^{2} )^{4} \)
19$C_2^2$ \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \)
23$C_2^2$ \( ( 1 + 238 T^{2} + p^{4} T^{4} )^{2} \)
29$C_2^2$ \( ( 1 + 710 T^{2} + p^{4} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 1438 T^{2} + p^{4} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \)
41$C_2$ \( ( 1 - 54 T + p^{2} T^{2} )^{4} \)
43$C_2^2$ \( ( 1 - 3266 T^{2} + p^{4} T^{4} )^{2} \)
47$C_2^2$ \( ( 1 - 3122 T^{2} + p^{4} T^{4} )^{2} \)
53$C_2^2$ \( ( 1 - 4474 T^{2} + p^{4} T^{4} )^{2} \)
59$C_2^2$ \( ( 1 - 3074 T^{2} + p^{4} T^{4} )^{2} \)
61$C_2$ \( ( 1 + p^{2} T^{2} )^{4} \)
67$C_2^2$ \( ( 1 - 5090 T^{2} + p^{4} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 1582 T^{2} + p^{4} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 - 9982 T^{2} + p^{4} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 13586 T^{2} + p^{4} T^{4} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p^{2} T^{2} )^{4} \)
97$C_2$ \( ( 1 + 34 T + p^{2} T^{2} )^{4} \)
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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

−7.10970251582771494239975284340, −7.05902065420215575671186662799, −6.88485258447142445661393499788, −6.32544178116755645472608079422, −6.26501213252446106558249353955, −5.92080424707621105397675523602, −5.89600092399411582189698518948, −5.80017860979144889507960188224, −5.33762162219964438338355147591, −5.15218647948462193728529203028, −4.52887528868130245609947294233, −4.40886813448565320613344482005, −4.38681170775438346408114038949, −4.13989780935152528447466377072, −3.80172373651684309210911486500, −3.71700161439641936047854552903, −3.10431082534423481500360114739, −2.64002327566505082916602582783, −2.47682631843069085085951303310, −2.30269252807935596139625529052, −2.08369824636502120831035129665, −1.81710504715945877767852023124, −0.957422685240773071435306177929, −0.73133241997383753962430576025, −0.094381138589818296010462128423, 0.094381138589818296010462128423, 0.73133241997383753962430576025, 0.957422685240773071435306177929, 1.81710504715945877767852023124, 2.08369824636502120831035129665, 2.30269252807935596139625529052, 2.47682631843069085085951303310, 2.64002327566505082916602582783, 3.10431082534423481500360114739, 3.71700161439641936047854552903, 3.80172373651684309210911486500, 4.13989780935152528447466377072, 4.38681170775438346408114038949, 4.40886813448565320613344482005, 4.52887528868130245609947294233, 5.15218647948462193728529203028, 5.33762162219964438338355147591, 5.80017860979144889507960188224, 5.89600092399411582189698518948, 5.92080424707621105397675523602, 6.26501213252446106558249353955, 6.32544178116755645472608079422, 6.88485258447142445661393499788, 7.05902065420215575671186662799, 7.10970251582771494239975284340

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