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Properties

Label 2-1008-63.58-c1-0-17
Degree 22
Conductor 10081008
Sign 0.0464−0.998i0.0464 - 0.998i
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  + (0.455 + 1.67i)3-s + (0.240 + 0.416i)5-s + (1.92 − 1.81i)7-s + (−2.58 + 1.52i)9-s + (1.69 − 2.92i)11-s + (−2.86 + 4.95i)13-s + (−0.587 + 0.592i)15-s + (2.75 + 4.77i)17-s + (−2.18 + 3.77i)19-s + (3.90 + 2.39i)21-s + (1.81 + 3.14i)23-s + (2.38 − 4.12i)25-s + (−3.71 − 3.62i)27-s + (1.53 + 2.65i)29-s + 9.34·31-s + ⋯
L(s)  = 1  + (0.262 + 0.964i)3-s + (0.107 + 0.186i)5-s + (0.728 − 0.684i)7-s + (−0.861 + 0.507i)9-s + (0.509 − 0.882i)11-s + (−0.793 + 1.37i)13-s + (−0.151 + 0.152i)15-s + (0.668 + 1.15i)17-s + (−0.500 + 0.866i)19-s + (0.852 + 0.522i)21-s + (0.378 + 0.654i)23-s + (0.476 − 0.825i)25-s + (−0.715 − 0.698i)27-s + (0.284 + 0.492i)29-s + 1.67·31-s + ⋯

Functional equation

Λ(s)=(1008s/2 ΓC(s) L(s)=((0.0464−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0464 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s+1/2) L(s)=((0.0464−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0464 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: 0.0464−0.998i0.0464 - 0.998i
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), 0.0464−0.998i)(2,\ 1008,\ (\ :1/2),\ 0.0464 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 1.8138956021.813895602
L(12)L(\frac12) ≈\approx 1.8138956021.813895602
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+(−0.455−1.67i)T 1 + (-0.455 - 1.67i)T
7 1+(−1.92+1.81i)T 1 + (-1.92 + 1.81i)T
good5 1+(−0.240−0.416i)T+(−2.5+4.33i)T2 1 + (-0.240 - 0.416i)T + (-2.5 + 4.33i)T^{2}
11 1+(−1.69+2.92i)T+(−5.5−9.52i)T2 1 + (-1.69 + 2.92i)T + (-5.5 - 9.52i)T^{2}
13 1+(2.86−4.95i)T+(−6.5−11.2i)T2 1 + (2.86 - 4.95i)T + (-6.5 - 11.2i)T^{2}
17 1+(−2.75−4.77i)T+(−8.5+14.7i)T2 1 + (-2.75 - 4.77i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.18−3.77i)T+(−9.5−16.4i)T2 1 + (2.18 - 3.77i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.81−3.14i)T+(−11.5+19.9i)T2 1 + (-1.81 - 3.14i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.53−2.65i)T+(−14.5+25.1i)T2 1 + (-1.53 - 2.65i)T + (-14.5 + 25.1i)T^{2}
31 1−9.34T+31T2 1 - 9.34T + 31T^{2}
37 1+(−1.48+2.57i)T+(−18.5−32.0i)T2 1 + (-1.48 + 2.57i)T + (-18.5 - 32.0i)T^{2}
41 1+(6.29−10.9i)T+(−20.5−35.5i)T2 1 + (6.29 - 10.9i)T + (-20.5 - 35.5i)T^{2}
43 1+(1.90+3.30i)T+(−21.5+37.2i)T2 1 + (1.90 + 3.30i)T + (-21.5 + 37.2i)T^{2}
47 1−3.76T+47T2 1 - 3.76T + 47T^{2}
53 1+(−5.57−9.66i)T+(−26.5+45.8i)T2 1 + (-5.57 - 9.66i)T + (-26.5 + 45.8i)T^{2}
59 1+8.42T+59T2 1 + 8.42T + 59T^{2}
61 1+7.28T+61T2 1 + 7.28T + 61T^{2}
67 1+2.57T+67T2 1 + 2.57T + 67T^{2}
71 1−3.94T+71T2 1 - 3.94T + 71T^{2}
73 1+(0.862+1.49i)T+(−36.5+63.2i)T2 1 + (0.862 + 1.49i)T + (-36.5 + 63.2i)T^{2}
79 1−5.59T+79T2 1 - 5.59T + 79T^{2}
83 1+(−0.119−0.206i)T+(−41.5+71.8i)T2 1 + (-0.119 - 0.206i)T + (-41.5 + 71.8i)T^{2}
89 1+(−0.648+1.12i)T+(−44.5−77.0i)T2 1 + (-0.648 + 1.12i)T + (-44.5 - 77.0i)T^{2}
97 1+(7.02+12.1i)T+(−48.5+84.0i)T2 1 + (7.02 + 12.1i)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

−10.26511759366407329704742720457, −9.404300681586833325543114908514, −8.498864110904275351096409724050, −7.940718784592278903785366981487, −6.71348062441072788540461140148, −5.84359195342085277354651272868, −4.63826935310822625773656127044, −4.09803833214536758597549854441, −3.02981543629401483000974866683, −1.56134398541828787041858788398, 0.865625624746330389148849504103, 2.26235885615271799936390518290, 2.99632622701526656872941174634, 4.77043760239972407306643435096, 5.34193224984671962494042682832, 6.52981983573093534418968042251, 7.32026676370057469313743825303, 8.039706021394538490868375877987, 8.822179763031672494140598270309, 9.595398280280161780827852934760

Graph of the ZZ-function along the critical line