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Properties

Label 2-1008-63.58-c1-0-39
Degree 22
Conductor 10081008
Sign 0.294+0.955i0.294 + 0.955i
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  + (1.65 + 0.523i)3-s + (−0.841 − 1.45i)5-s + (−1.65 − 2.06i)7-s + (2.45 + 1.72i)9-s + (0.622 − 1.07i)11-s + (1.96 − 3.39i)13-s + (−0.626 − 2.84i)15-s + (−1.62 − 2.81i)17-s + (−2.36 + 4.09i)19-s + (−1.65 − 4.27i)21-s + (−0.199 − 0.344i)23-s + (1.08 − 1.87i)25-s + (3.14 + 4.13i)27-s + (−3.19 − 5.54i)29-s + 0.578·31-s + ⋯
L(s)  = 1  + (0.953 + 0.302i)3-s + (−0.376 − 0.651i)5-s + (−0.625 − 0.780i)7-s + (0.817 + 0.576i)9-s + (0.187 − 0.325i)11-s + (0.543 − 0.941i)13-s + (−0.161 − 0.735i)15-s + (−0.394 − 0.683i)17-s + (−0.541 + 0.938i)19-s + (−0.360 − 0.932i)21-s + (−0.0415 − 0.0718i)23-s + (0.216 − 0.375i)25-s + (0.604 + 0.796i)27-s + (−0.594 − 1.02i)29-s + 0.103·31-s + ⋯

Functional equation

Λ(s)=(1008s/2 ΓC(s) L(s)=((0.294+0.955i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s+1/2) L(s)=((0.294+0.955i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: 0.294+0.955i0.294 + 0.955i
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), 0.294+0.955i)(2,\ 1008,\ (\ :1/2),\ 0.294 + 0.955i)

Particular Values

L(1)L(1) ≈\approx 1.8191686001.819168600
L(12)L(\frac12) ≈\approx 1.8191686001.819168600
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+(−1.65−0.523i)T 1 + (-1.65 - 0.523i)T
7 1+(1.65+2.06i)T 1 + (1.65 + 2.06i)T
good5 1+(0.841+1.45i)T+(−2.5+4.33i)T2 1 + (0.841 + 1.45i)T + (-2.5 + 4.33i)T^{2}
11 1+(−0.622+1.07i)T+(−5.5−9.52i)T2 1 + (-0.622 + 1.07i)T + (-5.5 - 9.52i)T^{2}
13 1+(−1.96+3.39i)T+(−6.5−11.2i)T2 1 + (-1.96 + 3.39i)T + (-6.5 - 11.2i)T^{2}
17 1+(1.62+2.81i)T+(−8.5+14.7i)T2 1 + (1.62 + 2.81i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.36−4.09i)T+(−9.5−16.4i)T2 1 + (2.36 - 4.09i)T + (-9.5 - 16.4i)T^{2}
23 1+(0.199+0.344i)T+(−11.5+19.9i)T2 1 + (0.199 + 0.344i)T + (-11.5 + 19.9i)T^{2}
29 1+(3.19+5.54i)T+(−14.5+25.1i)T2 1 + (3.19 + 5.54i)T + (-14.5 + 25.1i)T^{2}
31 1−0.578T+31T2 1 - 0.578T + 31T^{2}
37 1+(−2.72+4.71i)T+(−18.5−32.0i)T2 1 + (-2.72 + 4.71i)T + (-18.5 - 32.0i)T^{2}
41 1+(−4.20+7.27i)T+(−20.5−35.5i)T2 1 + (-4.20 + 7.27i)T + (-20.5 - 35.5i)T^{2}
43 1+(2.46+4.26i)T+(−21.5+37.2i)T2 1 + (2.46 + 4.26i)T + (-21.5 + 37.2i)T^{2}
47 1−0.425T+47T2 1 - 0.425T + 47T^{2}
53 1+(0.466+0.807i)T+(−26.5+45.8i)T2 1 + (0.466 + 0.807i)T + (-26.5 + 45.8i)T^{2}
59 1+6.05T+59T2 1 + 6.05T + 59T^{2}
61 1−10.2T+61T2 1 - 10.2T + 61T^{2}
67 1−9.41T+67T2 1 - 9.41T + 67T^{2}
71 1+8.46T+71T2 1 + 8.46T + 71T^{2}
73 1+(−6.82−11.8i)T+(−36.5+63.2i)T2 1 + (-6.82 - 11.8i)T + (-36.5 + 63.2i)T^{2}
79 1−5.53T+79T2 1 - 5.53T + 79T^{2}
83 1+(−8.03−13.9i)T+(−41.5+71.8i)T2 1 + (-8.03 - 13.9i)T + (-41.5 + 71.8i)T^{2}
89 1+(6.03−10.4i)T+(−44.5−77.0i)T2 1 + (6.03 - 10.4i)T + (-44.5 - 77.0i)T^{2}
97 1+(5.86+10.1i)T+(−48.5+84.0i)T2 1 + (5.86 + 10.1i)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

−9.740902613154469337156284394711, −8.914269941538164361099730955157, −8.183345473213814923090382406114, −7.55087930098286333910318775843, −6.50242785489621386845902393844, −5.35284194845050011067065628181, −4.10660791421427195389284238942, −3.70258499917188875707760265364, −2.43276302136549163859671483709, −0.75508256751896699185577131204, 1.75401866983436407903183279035, 2.82466862415444116067458929697, 3.64159604929253198430355184744, 4.67177886367156031860706751673, 6.29713443057348450164217093368, 6.70596029091311366504029906405, 7.61733036382599719782757000518, 8.658095905773061853296731383744, 9.116146796148702541377311418105, 9.918319363017319730932483752496

Graph of the ZZ-function along the critical line