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Properties

Label 2-1008-63.58-c1-0-12
Degree 22
Conductor 10081008
Sign 0.964+0.262i0.964 + 0.262i
Analytic cond. 8.048928.04892
Root an. cond. 2.837062.83706
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.12 − 1.31i)3-s + (−0.927 − 1.60i)5-s + (−0.900 + 2.48i)7-s + (−0.467 + 2.96i)9-s + (−1.28 + 2.23i)11-s + (2.82 − 4.88i)13-s + (−1.07 + 3.03i)15-s + (3.57 + 6.19i)17-s + (−0.636 + 1.10i)19-s + (4.28 − 1.61i)21-s + (0.120 + 0.208i)23-s + (0.777 − 1.34i)25-s + (4.42 − 2.71i)27-s + (0.923 + 1.59i)29-s + 2.99·31-s + ⋯
L(s)  = 1  + (−0.649 − 0.760i)3-s + (−0.414 − 0.718i)5-s + (−0.340 + 0.940i)7-s + (−0.155 + 0.987i)9-s + (−0.388 + 0.672i)11-s + (0.782 − 1.35i)13-s + (−0.276 + 0.782i)15-s + (0.868 + 1.50i)17-s + (−0.146 + 0.252i)19-s + (0.935 − 0.352i)21-s + (0.0251 + 0.0435i)23-s + (0.155 − 0.269i)25-s + (0.852 − 0.523i)27-s + (0.171 + 0.297i)29-s + 0.537·31-s + ⋯

Functional equation

Λ(s)=(1008s/2 ΓC(s) L(s)=((0.964+0.262i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s+1/2) L(s)=((0.964+0.262i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: 0.964+0.262i0.964 + 0.262i
Analytic conductor: 8.048928.04892
Root analytic conductor: 2.837062.83706
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1008(625,⋅)\chi_{1008} (625, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1008, ( :1/2), 0.964+0.262i)(2,\ 1008,\ (\ :1/2),\ 0.964 + 0.262i)

Particular Values

L(1)L(1) ≈\approx 1.0727327641.072732764
L(12)L(\frac12) ≈\approx 1.0727327641.072732764
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}
ppFp(T)F_p(T)
bad2 1 1
3 1+(1.12+1.31i)T 1 + (1.12 + 1.31i)T
7 1+(0.900−2.48i)T 1 + (0.900 - 2.48i)T
good5 1+(0.927+1.60i)T+(−2.5+4.33i)T2 1 + (0.927 + 1.60i)T + (-2.5 + 4.33i)T^{2}
11 1+(1.28−2.23i)T+(−5.5−9.52i)T2 1 + (1.28 - 2.23i)T + (-5.5 - 9.52i)T^{2}
13 1+(−2.82+4.88i)T+(−6.5−11.2i)T2 1 + (-2.82 + 4.88i)T + (-6.5 - 11.2i)T^{2}
17 1+(−3.57−6.19i)T+(−8.5+14.7i)T2 1 + (-3.57 - 6.19i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.636−1.10i)T+(−9.5−16.4i)T2 1 + (0.636 - 1.10i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.120−0.208i)T+(−11.5+19.9i)T2 1 + (-0.120 - 0.208i)T + (-11.5 + 19.9i)T^{2}
29 1+(−0.923−1.59i)T+(−14.5+25.1i)T2 1 + (-0.923 - 1.59i)T + (-14.5 + 25.1i)T^{2}
31 1−2.99T+31T2 1 - 2.99T + 31T^{2}
37 1+(−0.338+0.585i)T+(−18.5−32.0i)T2 1 + (-0.338 + 0.585i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.733−1.27i)T+(−20.5−35.5i)T2 1 + (0.733 - 1.27i)T + (-20.5 - 35.5i)T^{2}
43 1+(4.14+7.17i)T+(−21.5+37.2i)T2 1 + (4.14 + 7.17i)T + (-21.5 + 37.2i)T^{2}
47 1−12.3T+47T2 1 - 12.3T + 47T^{2}
53 1+(−3.35−5.81i)T+(−26.5+45.8i)T2 1 + (-3.35 - 5.81i)T + (-26.5 + 45.8i)T^{2}
59 1+2.08T+59T2 1 + 2.08T + 59T^{2}
61 1−12.9T+61T2 1 - 12.9T + 61T^{2}
67 1−4.83T+67T2 1 - 4.83T + 67T^{2}
71 1−1.53T+71T2 1 - 1.53T + 71T^{2}
73 1+(6.55+11.3i)T+(−36.5+63.2i)T2 1 + (6.55 + 11.3i)T + (-36.5 + 63.2i)T^{2}
79 1−3.72T+79T2 1 - 3.72T + 79T^{2}
83 1+(−3.00−5.19i)T+(−41.5+71.8i)T2 1 + (-3.00 - 5.19i)T + (-41.5 + 71.8i)T^{2}
89 1+(−6.60+11.4i)T+(−44.5−77.0i)T2 1 + (-6.60 + 11.4i)T + (-44.5 - 77.0i)T^{2}
97 1+(−6.40−11.1i)T+(−48.5+84.0i)T2 1 + (-6.40 - 11.1i)T + (-48.5 + 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.23070812237892164639764897211, −8.788919534544587309139373230099, −8.236338197105008065509163206762, −7.57146109022385378779603514328, −6.32144662689006173621269742320, −5.69422413997251172668490863086, −4.97341887483639272168638670029, −3.62407396109202787745775841484, −2.27214822539353163513701939871, −0.934849982196174165786965915114, 0.75806167273809316065007213949, 2.98088058270800704184233906465, 3.76774369150015926409338146360, 4.62110420260357015130031112208, 5.71111645069640655938642506160, 6.71839190780547668043738630972, 7.18435727380486293873498017040, 8.413018492406314468682201770666, 9.430208859531889475971648029384, 10.05456265511147264236107887122

Graph of the ZZ-function along the critical line