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Properties

Label 2-1008-63.59-c1-0-26
Degree 22
Conductor 10081008
Sign 0.861−0.507i0.861 - 0.507i
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  + (−0.626 + 1.61i)3-s + 3.55·5-s + (1.49 − 2.18i)7-s + (−2.21 − 2.02i)9-s + 3.02i·11-s + (0.888 + 0.513i)13-s + (−2.22 + 5.73i)15-s + (0.809 − 1.40i)17-s + (7.12 − 4.11i)19-s + (2.58 + 3.78i)21-s − 3.35i·23-s + 7.61·25-s + (4.65 − 2.30i)27-s + (−3.70 + 2.13i)29-s + (−5.18 + 2.99i)31-s + ⋯
L(s)  = 1  + (−0.361 + 0.932i)3-s + 1.58·5-s + (0.564 − 0.825i)7-s + (−0.738 − 0.674i)9-s + 0.911i·11-s + (0.246 + 0.142i)13-s + (−0.574 + 1.48i)15-s + (0.196 − 0.339i)17-s + (1.63 − 0.943i)19-s + (0.565 + 0.824i)21-s − 0.700i·23-s + 1.52·25-s + (0.895 − 0.444i)27-s + (−0.687 + 0.397i)29-s + (−0.931 + 0.537i)31-s + ⋯

Functional equation

Λ(s)=(1008s/2 ΓC(s) L(s)=((0.861−0.507i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s+1/2) L(s)=((0.861−0.507i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: 0.861−0.507i0.861 - 0.507i
Analytic conductor: 8.048928.04892
Root analytic conductor: 2.837062.83706
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1008(689,⋅)\chi_{1008} (689, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1008, ( :1/2), 0.861−0.507i)(2,\ 1008,\ (\ :1/2),\ 0.861 - 0.507i)

Particular Values

L(1)L(1) ≈\approx 2.0366227622.036622762
L(12)L(\frac12) ≈\approx 2.0366227622.036622762
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+(0.626−1.61i)T 1 + (0.626 - 1.61i)T
7 1+(−1.49+2.18i)T 1 + (-1.49 + 2.18i)T
good5 1−3.55T+5T2 1 - 3.55T + 5T^{2}
11 1−3.02iT−11T2 1 - 3.02iT - 11T^{2}
13 1+(−0.888−0.513i)T+(6.5+11.2i)T2 1 + (-0.888 - 0.513i)T + (6.5 + 11.2i)T^{2}
17 1+(−0.809+1.40i)T+(−8.5−14.7i)T2 1 + (-0.809 + 1.40i)T + (-8.5 - 14.7i)T^{2}
19 1+(−7.12+4.11i)T+(9.5−16.4i)T2 1 + (-7.12 + 4.11i)T + (9.5 - 16.4i)T^{2}
23 1+3.35iT−23T2 1 + 3.35iT - 23T^{2}
29 1+(3.70−2.13i)T+(14.5−25.1i)T2 1 + (3.70 - 2.13i)T + (14.5 - 25.1i)T^{2}
31 1+(5.18−2.99i)T+(15.5−26.8i)T2 1 + (5.18 - 2.99i)T + (15.5 - 26.8i)T^{2}
37 1+(−2.92−5.06i)T+(−18.5+32.0i)T2 1 + (-2.92 - 5.06i)T + (-18.5 + 32.0i)T^{2}
41 1+(0.0472−0.0817i)T+(−20.5−35.5i)T2 1 + (0.0472 - 0.0817i)T + (-20.5 - 35.5i)T^{2}
43 1+(3.05+5.29i)T+(−21.5+37.2i)T2 1 + (3.05 + 5.29i)T + (-21.5 + 37.2i)T^{2}
47 1+(−2.57+4.45i)T+(−23.5−40.7i)T2 1 + (-2.57 + 4.45i)T + (-23.5 - 40.7i)T^{2}
53 1+(−2.76−1.59i)T+(26.5+45.8i)T2 1 + (-2.76 - 1.59i)T + (26.5 + 45.8i)T^{2}
59 1+(−4.42−7.65i)T+(−29.5+51.0i)T2 1 + (-4.42 - 7.65i)T + (-29.5 + 51.0i)T^{2}
61 1+(4.06+2.34i)T+(30.5+52.8i)T2 1 + (4.06 + 2.34i)T + (30.5 + 52.8i)T^{2}
67 1+(0.187+0.325i)T+(−33.5+58.0i)T2 1 + (0.187 + 0.325i)T + (-33.5 + 58.0i)T^{2}
71 1−13.9iT−71T2 1 - 13.9iT - 71T^{2}
73 1+(−1.13−0.655i)T+(36.5+63.2i)T2 1 + (-1.13 - 0.655i)T + (36.5 + 63.2i)T^{2}
79 1+(−0.462+0.800i)T+(−39.5−68.4i)T2 1 + (-0.462 + 0.800i)T + (-39.5 - 68.4i)T^{2}
83 1+(−5.43−9.40i)T+(−41.5+71.8i)T2 1 + (-5.43 - 9.40i)T + (-41.5 + 71.8i)T^{2}
89 1+(−2.35−4.07i)T+(−44.5+77.0i)T2 1 + (-2.35 - 4.07i)T + (-44.5 + 77.0i)T^{2}
97 1+(13.3−7.69i)T+(48.5−84.0i)T2 1 + (13.3 - 7.69i)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

−9.970531546865623461832351475885, −9.485775627572941408939272213876, −8.718380331813144041113136169569, −7.31380088536268794857238753859, −6.61764437472785222142549869416, −5.35632388466856561355816537011, −5.08926485825118211695136217192, −3.93011918137925250048204219136, −2.63034892983714063230346651007, −1.25976103730187911795230654962, 1.31041207747980465225106562648, 2.10044019997712020117493553285, 3.25727086151644181599991334074, 5.20674788385239550242618139238, 5.82758001471901384647835643448, 6.04693906362197400018403753152, 7.43643440257349938449637387201, 8.151216642740362321773645640024, 9.132408062495751576379362171697, 9.750625620028818004796342152567

Graph of the ZZ-function along the critical line