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Properties

Label 8-1280e4-1.1-c1e4-0-12
Degree 88
Conductor 2.684×10122.684\times 10^{12}
Sign 11
Analytic cond. 10913.110913.1
Root an. cond. 3.197003.19700
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s + 4·13-s + 4·17-s + 38·25-s + 16·29-s + 20·37-s − 8·41-s + 28·53-s − 32·65-s + 28·73-s + 18·81-s − 32·85-s − 28·97-s + 48·109-s + 36·113-s − 20·121-s − 136·125-s + 127-s + 131-s + 137-s + 139-s − 128·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 3.57·5-s + 1.10·13-s + 0.970·17-s + 38/5·25-s + 2.97·29-s + 3.28·37-s − 1.24·41-s + 3.84·53-s − 3.96·65-s + 3.27·73-s + 2·81-s − 3.47·85-s − 2.84·97-s + 4.59·109-s + 3.38·113-s − 1.81·121-s − 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.6·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

Λ(s)=((232⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((232⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 232⋅542^{32} \cdot 5^{4}
Sign: 11
Analytic conductor: 10913.110913.1
Root analytic conductor: 3.197003.19700
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 232⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{32} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.4609274372.460927437
L(12)L(\frac12) ≈\approx 2.4609274372.460927437
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)
bad2 1 1
5C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
good3C2C_2×\timesC2C_2 (1−pT2)2(1+pT2)2 ( 1 - p T^{2} )^{2}( 1 + p T^{2} )^{2}
7C23C_2^3 1−34T4+p4T8 1 - 34 T^{4} + p^{4} T^{8}
11C22C_2^2 (1+10T2+p2T4)2 ( 1 + 10 T^{2} + p^{2} T^{4} )^{2}
13C2C_2 (1−6T+pT2)2(1+4T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}
17C22C_2^2 (1−2T+2T2−2pT3+p2T4)2 ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}
19C22C_2^2 (1+10T2+p2T4)2 ( 1 + 10 T^{2} + p^{2} T^{4} )^{2}
23C23C_2^3 1+542T4+p4T8 1 + 542 T^{4} + p^{4} T^{8}
29C2C_2 (1−4T+pT2)4 ( 1 - 4 T + p T^{2} )^{4}
31C22C_2^2 (1−50T2+p2T4)2 ( 1 - 50 T^{2} + p^{2} T^{4} )^{2}
37C2C_2 (1−12T+pT2)2(1+2T+pT2)2 ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2}
41C2C_2 (1+2T+pT2)4 ( 1 + 2 T + p T^{2} )^{4}
43C23C_2^3 1+2702T4+p4T8 1 + 2702 T^{4} + p^{4} T^{8}
47C23C_2^3 1+3326T4+p4T8 1 + 3326 T^{4} + p^{4} T^{8}
53C22C_2^2 (1−14T+98T2−14pT3+p2T4)2 ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}
59C22C_2^2 (1−70T2+p2T4)2 ( 1 - 70 T^{2} + p^{2} T^{4} )^{2}
61C22C_2^2 (1−86T2+p2T4)2 ( 1 - 86 T^{2} + p^{2} T^{4} )^{2}
67C23C_2^3 1−2578T4+p4T8 1 - 2578 T^{4} + p^{4} T^{8}
71C22C_2^2 (1−34T2+p2T4)2 ( 1 - 34 T^{2} + p^{2} T^{4} )^{2}
73C22C_2^2 (1−14T+98T2−14pT3+p2T4)2 ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}
79C2C_2 (1+pT2)4 ( 1 + p T^{2} )^{4}
83C23C_2^3 1+2606T4+p4T8 1 + 2606 T^{4} + p^{4} T^{8}
89C22C_2^2 (1−114T2+p2T4)2 ( 1 - 114 T^{2} + p^{2} T^{4} )^{2}
97C22C_2^2 (1+14T+98T2+14pT3+p2T4)2 ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + 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

−7.20924957123185824353214197929, −6.58325554751861152459748669912, −6.54924465860822829057565795998, −6.34242811483932212847056428215, −6.20046281201546766898151929982, −5.87858510084929993875931566238, −5.49417876278610238580917008599, −5.22749700962056678580572602532, −4.99218500360125379989454933050, −4.82787669480671824358480454860, −4.58329646297923248248155734147, −4.29358123229920425813289483807, −4.15252359049748957888910733822, −3.93090077919749968533425366975, −3.55991014931965785906009800304, −3.52288164616723216047463528685, −3.42233251412544790252163464322, −2.86512113557611190551058533997, −2.67356148997737691957147354396, −2.53742082632452432688807137161, −2.05601865727651715919998004771, −1.31482852011629183228314243358, −0.844412216663648797129275997289, −0.830057158570300643312637949553, −0.57071229291142128264431326086, 0.57071229291142128264431326086, 0.830057158570300643312637949553, 0.844412216663648797129275997289, 1.31482852011629183228314243358, 2.05601865727651715919998004771, 2.53742082632452432688807137161, 2.67356148997737691957147354396, 2.86512113557611190551058533997, 3.42233251412544790252163464322, 3.52288164616723216047463528685, 3.55991014931965785906009800304, 3.93090077919749968533425366975, 4.15252359049748957888910733822, 4.29358123229920425813289483807, 4.58329646297923248248155734147, 4.82787669480671824358480454860, 4.99218500360125379989454933050, 5.22749700962056678580572602532, 5.49417876278610238580917008599, 5.87858510084929993875931566238, 6.20046281201546766898151929982, 6.34242811483932212847056428215, 6.54924465860822829057565795998, 6.58325554751861152459748669912, 7.20924957123185824353214197929

Graph of the ZZ-function along the critical line