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Properties

Label 8-45e4-1.1-c5e4-0-1
Degree 88
Conductor 41006254100625
Sign 11
Analytic cond. 2713.262713.26
Root an. cond. 2.686492.68649
Motivic weight 55
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  + 88·4-s + 3.76e3·16-s − 1.93e3·19-s + 3.83e3·25-s + 1.44e4·31-s − 3.35e4·49-s + 8.54e4·61-s + 7.04e4·64-s − 1.70e5·76-s − 3.98e5·79-s + 3.37e5·100-s + 6.77e3·109-s − 1.40e5·121-s + 1.27e6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.38e6·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 11/4·4-s + 3.67·16-s − 1.23·19-s + 1.22·25-s + 2.69·31-s − 1.99·49-s + 2.94·61-s + 2.14·64-s − 3.38·76-s − 7.18·79-s + 3.37·100-s + 0.0546·109-s − 0.870·121-s + 7.41·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 3.72·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + ⋯

Functional equation

Λ(s)=(4100625s/2 ΓC(s)4 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(4100625s/2 ΓC(s+5/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 41006254100625    =    38⋅543^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 2713.262713.26
Root analytic conductor: 2.686492.68649
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 4100625, ( :5/2,5/2,5/2,5/2), 1)(8,\ 4100625,\ (\ :5/2, 5/2, 5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 6.9097294366.909729436
L(12)L(\frac12) ≈\approx 6.9097294366.909729436
L(72)L(\frac{7}{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)
bad3 1 1
5C22C_2^2 1−766pT2+p10T4 1 - 766 p T^{2} + p^{10} T^{4}
good2C22C_2^2 (1−11p2T2+p10T4)2 ( 1 - 11 p^{2} T^{2} + p^{10} T^{4} )^{2}
7C22C_2^2 (1+2398pT2+p10T4)2 ( 1 + 2398 p T^{2} + p^{10} T^{4} )^{2}
11C22C_2^2 (1+70102T2+p10T4)2 ( 1 + 70102 T^{2} + p^{10} T^{4} )^{2}
13C22C_2^2 (1−692186T2+p10T4)2 ( 1 - 692186 T^{2} + p^{10} T^{4} )^{2}
17C22C_2^2 (1−57134T2+p10T4)2 ( 1 - 57134 T^{2} + p^{10} T^{4} )^{2}
19C2C_2 (1+484T+p5T2)4 ( 1 + 484 T + p^{5} T^{2} )^{4}
23C22C_2^2 (1−14654p2T2+p10T4)2 ( 1 - 14654 p^{2} T^{2} + p^{10} T^{4} )^{2}
29C22C_2^2 (1+10530298T2+p10T4)2 ( 1 + 10530298 T^{2} + p^{10} T^{4} )^{2}
31C2C_2 (1−3608T+p5T2)4 ( 1 - 3608 T + p^{5} T^{2} )^{4}
37C22C_2^2 (1−83802314T2+p10T4)2 ( 1 - 83802314 T^{2} + p^{10} T^{4} )^{2}
41C22C_2^2 (1+109744402T2+p10T4)2 ( 1 + 109744402 T^{2} + p^{10} T^{4} )^{2}
43C22C_2^2 (1−135962486T2+p10T4)2 ( 1 - 135962486 T^{2} + p^{10} T^{4} )^{2}
47C22C_2^2 (1−367610894T2+p10T4)2 ( 1 - 367610894 T^{2} + p^{10} T^{4} )^{2}
53C22C_2^2 (1−813621206T2+p10T4)2 ( 1 - 813621206 T^{2} + p^{10} T^{4} )^{2}
59C22C_2^2 (1+1399356598T2+p10T4)2 ( 1 + 1399356598 T^{2} + p^{10} T^{4} )^{2}
61C2C_2 (1−21362T+p5T2)4 ( 1 - 21362 T + p^{5} T^{2} )^{4}
67C22C_2^2 (1−1504963814T2+p10T4)2 ( 1 - 1504963814 T^{2} + p^{10} T^{4} )^{2}
71C22C_2^2 (1+2510746702T2+p10T4)2 ( 1 + 2510746702 T^{2} + p^{10} T^{4} )^{2}
73C22C_2^2 (1−4138885586T2+p10T4)2 ( 1 - 4138885586 T^{2} + p^{10} T^{4} )^{2}
79C2C_2 (1+99616T+p5T2)4 ( 1 + 99616 T + p^{5} T^{2} )^{4}
83C22C_2^2 (1−4615601606T2+p10T4)2 ( 1 - 4615601606 T^{2} + p^{10} T^{4} )^{2}
89C22C_2^2 (1−3347081102T2+p10T4)2 ( 1 - 3347081102 T^{2} + p^{10} T^{4} )^{2}
97C22C_2^2 (1−13052162114T2+p10T4)2 ( 1 - 13052162114 T^{2} + p^{10} 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

−10.83224676582407332869034412173, −10.66341554480011279125419100584, −10.06257577487544824513806148999, −9.994062621061679483891007575304, −9.875688674686054121279194008582, −9.038678790073071448155169786653, −8.549920576038356203551742193463, −8.511137356933303855154340509177, −8.038277402242260252965844974223, −7.70530314059212406696192150973, −7.02392205339127744933052634002, −6.94863621667277357273183464548, −6.82059201830496534405408186100, −6.24346348423030527136328636253, −6.12352382054055081290956472604, −5.60942258597415453355976194803, −5.06591553333735791787709444121, −4.32768687810259873268172939927, −4.23536654593675827458714478007, −3.04446326279113337484234757229, −2.93966998746581877814460196130, −2.57913545993673695142714321798, −1.82450157580525235902138059426, −1.51537898602479532329673229826, −0.61890446605563412164653789820, 0.61890446605563412164653789820, 1.51537898602479532329673229826, 1.82450157580525235902138059426, 2.57913545993673695142714321798, 2.93966998746581877814460196130, 3.04446326279113337484234757229, 4.23536654593675827458714478007, 4.32768687810259873268172939927, 5.06591553333735791787709444121, 5.60942258597415453355976194803, 6.12352382054055081290956472604, 6.24346348423030527136328636253, 6.82059201830496534405408186100, 6.94863621667277357273183464548, 7.02392205339127744933052634002, 7.70530314059212406696192150973, 8.038277402242260252965844974223, 8.511137356933303855154340509177, 8.549920576038356203551742193463, 9.038678790073071448155169786653, 9.875688674686054121279194008582, 9.994062621061679483891007575304, 10.06257577487544824513806148999, 10.66341554480011279125419100584, 10.83224676582407332869034412173

Graph of the ZZ-function along the critical line