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Properties

Label 8-3700e4-1.1-c0e4-0-3
Degree 88
Conductor 1.874×10141.874\times 10^{14}
Sign 11
Analytic cond. 11.626111.6261
Root an. cond. 1.358871.35887
Motivic weight 00
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 2·17-s + 2·29-s + 6·41-s + 2·53-s − 4·61-s − 64-s − 2·68-s + 4·73-s + 81-s − 2·89-s + 2·109-s + 4·113-s + 2·116-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 6·164-s + 167-s − 2·169-s + 173-s + ⋯
L(s)  = 1  + 4-s − 2·17-s + 2·29-s + 6·41-s + 2·53-s − 4·61-s − 64-s − 2·68-s + 4·73-s + 81-s − 2·89-s + 2·109-s + 4·113-s + 2·116-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 6·164-s + 167-s − 2·169-s + 173-s + ⋯

Functional equation

Λ(s)=((28⋅58⋅374)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}
Λ(s)=((28⋅58⋅374)s/2 ΓC(s)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 28⋅58⋅3742^{8} \cdot 5^{8} \cdot 37^{4}
Sign: 11
Analytic conductor: 11.626111.6261
Root analytic conductor: 1.358871.35887
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 28⋅58⋅374, ( :0,0,0,0), 1)(8,\ 2^{8} \cdot 5^{8} \cdot 37^{4} ,\ ( \ : 0, 0, 0, 0 ),\ 1 )

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.4833252292.483325229
L(12)L(\frac12) ≈\approx 2.4833252292.483325229
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
5 1 1
37C22C_2^2 1−T2+T4 1 - T^{2} + T^{4}
good3C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
7C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
11C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
13C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
17C1C_1×\timesC2C_2 (1+T)4(1−T+T2)2 ( 1 + T )^{4}( 1 - T + T^{2} )^{2}
19C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
23C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
29C2C_2×\timesC22C_2^2 (1−T+T2)2(1−T2+T4) ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )
31C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
41C1C_1×\timesC2C_2 (1−T)4(1−T+T2)2 ( 1 - T )^{4}( 1 - T + T^{2} )^{2}
43C2C_2 (1+T2)4 ( 1 + T^{2} )^{4}
47C22C_2^2 (1+T4)2 ( 1 + T^{4} )^{2}
53C2C_2×\timesC22C_2^2 (1−T+T2)2(1−T2+T4) ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} )
59C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
61C1C_1×\timesC22C_2^2 (1+T)4(1−T2+T4) ( 1 + T )^{4}( 1 - T^{2} + T^{4} )
67C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
71C2C_2 (1−T+T2)2(1+T+T2)2 ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2}
73C1C_1×\timesC2C_2 (1−T)4(1+T2)2 ( 1 - T )^{4}( 1 + T^{2} )^{2}
79C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
83C23C_2^3 1−T4+T8 1 - T^{4} + T^{8}
89C2C_2×\timesC2C_2 (1+T2)2(1+T+T2)2 ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2}
97C22C_2^2 (1−T2+T4)2 ( 1 - T^{2} + T^{4} )^{2}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(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

−6.16630630524096794131161258355, −6.14471679369044540232954315180, −5.88631361975778330899983581780, −5.85914437662118805665244560444, −5.42308971380183987536545571423, −5.14000427105691417025556327076, −5.11308147018239799802673424525, −4.63403413753255771279975077233, −4.54423795827166947729060764740, −4.51213308020335398300577713798, −4.28419056230916022995080388534, −3.97613326194533055134601868448, −3.76578334414380738175481936607, −3.73479125326380864409426117049, −3.04665899878209660345667356242, −3.04606007709590653142412796751, −2.81413482607366074624121339362, −2.58480761748897134067001321638, −2.38937685184932740618524699464, −2.12913942530925576557983242255, −2.03084534267029494748203269663, −1.71114282316159398577028843638, −1.06564558925585274278139865091, −1.01569777304532563367244706446, −0.63715488049539210376889932840, 0.63715488049539210376889932840, 1.01569777304532563367244706446, 1.06564558925585274278139865091, 1.71114282316159398577028843638, 2.03084534267029494748203269663, 2.12913942530925576557983242255, 2.38937685184932740618524699464, 2.58480761748897134067001321638, 2.81413482607366074624121339362, 3.04606007709590653142412796751, 3.04665899878209660345667356242, 3.73479125326380864409426117049, 3.76578334414380738175481936607, 3.97613326194533055134601868448, 4.28419056230916022995080388534, 4.51213308020335398300577713798, 4.54423795827166947729060764740, 4.63403413753255771279975077233, 5.11308147018239799802673424525, 5.14000427105691417025556327076, 5.42308971380183987536545571423, 5.85914437662118805665244560444, 5.88631361975778330899983581780, 6.14471679369044540232954315180, 6.16630630524096794131161258355

Graph of the ZZ-function along the critical line