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Properties

Label 8-525e4-1.1-c1e4-0-6
Degree 88
Conductor 7596914062575969140625
Sign 11
Analytic cond. 308.848308.848
Root an. cond. 2.047472.04747
Motivic weight 11
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  − 8·4-s + 6·9-s + 40·16-s − 48·36-s − 11·49-s − 160·64-s + 16·79-s + 27·81-s − 76·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 240·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 88·196-s + ⋯
L(s)  = 1  − 4·4-s + 2·9-s + 10·16-s − 8·36-s − 1.57·49-s − 20·64-s + 1.80·79-s + 3·81-s − 7.27·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 20·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 44/7·196-s + ⋯

Functional equation

Λ(s)=((34⋅58⋅74)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((34⋅58⋅74)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 34⋅58⋅743^{4} \cdot 5^{8} \cdot 7^{4}
Sign: 11
Analytic conductor: 308.848308.848
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 34⋅58⋅74, ( :1/2,1/2,1/2,1/2), 1)(8,\ 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.51396875870.5139687587
L(12)L(\frac12) ≈\approx 0.51396875870.5139687587
L(32)L(\frac{3}{2}) 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)
bad3C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
5 1 1
7C22C_2^2 1+11T2+p2T4 1 + 11 T^{2} + p^{2} T^{4}
good2C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
11C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
13C22C_2^2 (1−T2+p2T4)2 ( 1 - T^{2} + p^{2} T^{4} )^{2}
17C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
19C2C_2 (1−T+pT2)2(1+T+pT2)2 ( 1 - T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2}
23C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
29C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
31C2C_2 (1−7T+pT2)2(1+7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2}
37C22C_2^2 (1+26T2+p2T4)2 ( 1 + 26 T^{2} + p^{2} T^{4} )^{2}
41C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
43C22C_2^2 (1−61T2+p2T4)2 ( 1 - 61 T^{2} + p^{2} T^{4} )^{2}
47C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
53C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
59C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
61C2C_2 (1−13T+pT2)2(1+13T+pT2)2 ( 1 - 13 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2}
67C22C_2^2 (1−109T2+p2T4)2 ( 1 - 109 T^{2} + p^{2} T^{4} )^{2}
71C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
73C22C_2^2 (1−46T2+p2T4)2 ( 1 - 46 T^{2} + p^{2} T^{4} )^{2}
79C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
83C2C_2 (1−pT2)4 ( 1 - p T^{2} )^{4}
89C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
97C22C_2^2 (1−169T2+p2T4)2 ( 1 - 169 T^{2} + p^{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

−7.892373657994132158383364893483, −7.72472803048816042952177769385, −7.52691468897359460854277318247, −7.31909911075668038044554405328, −6.94528768279055183720331696645, −6.48180853456505477535957617944, −6.27331866249875387612168106742, −6.26650531315286869363913951717, −5.65537697477364257038411022252, −5.34934078652998875859280284817, −5.25319537156480419360196590242, −5.05521707132237474578327744846, −4.69970713442226506322571479400, −4.65708942210780292188596659207, −4.23950650292507199390242448515, −3.97788750817692412371426635244, −3.95588915906764318521666195230, −3.63816324703329082798230012468, −3.30527053522407803150577570583, −2.98904242763748093802211547049, −2.40628662692044119453696030807, −1.65346617475158190382470573438, −1.32405775916972774205726664311, −1.02685279558057878079671248462, −0.33090966372630186139671030315, 0.33090966372630186139671030315, 1.02685279558057878079671248462, 1.32405775916972774205726664311, 1.65346617475158190382470573438, 2.40628662692044119453696030807, 2.98904242763748093802211547049, 3.30527053522407803150577570583, 3.63816324703329082798230012468, 3.95588915906764318521666195230, 3.97788750817692412371426635244, 4.23950650292507199390242448515, 4.65708942210780292188596659207, 4.69970713442226506322571479400, 5.05521707132237474578327744846, 5.25319537156480419360196590242, 5.34934078652998875859280284817, 5.65537697477364257038411022252, 6.26650531315286869363913951717, 6.27331866249875387612168106742, 6.48180853456505477535957617944, 6.94528768279055183720331696645, 7.31909911075668038044554405328, 7.52691468897359460854277318247, 7.72472803048816042952177769385, 7.892373657994132158383364893483

Graph of the ZZ-function along the critical line