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Properties

Label 2-2646-63.16-c1-0-33
Degree 22
Conductor 26462646
Sign −0.959+0.282i-0.959 + 0.282i
Analytic cond. 21.128421.1284
Root an. cond. 4.596564.59656
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.517·5-s + 0.999·8-s + (0.258 + 0.448i)10-s + 1.46·11-s + (1.22 + 2.12i)13-s + (−0.5 − 0.866i)16-s + (−1.74 − 3.01i)17-s + (−0.258 + 0.448i)19-s + (0.258 − 0.448i)20-s + (−0.732 − 1.26i)22-s − 7.92·23-s − 4.73·25-s + (1.22 − 2.12i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.231·5-s + 0.353·8-s + (0.0818 + 0.141i)10-s + 0.441·11-s + (0.339 + 0.588i)13-s + (−0.125 − 0.216i)16-s + (−0.422 − 0.731i)17-s + (−0.0593 + 0.102i)19-s + (0.0578 − 0.100i)20-s + (−0.156 − 0.270i)22-s − 1.65·23-s − 0.946·25-s + (0.240 − 0.416i)26-s + ⋯

Functional equation

Λ(s)=(2646s/2 ΓC(s) L(s)=((−0.959+0.282i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2646s/2 ΓC(s+1/2) L(s)=((−0.959+0.282i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26462646    =    2⋅33⋅722 \cdot 3^{3} \cdot 7^{2}
Sign: −0.959+0.282i-0.959 + 0.282i
Analytic conductor: 21.128421.1284
Root analytic conductor: 4.596564.59656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2646(667,⋅)\chi_{2646} (667, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2646, ( :1/2), −0.959+0.282i)(2,\ 2646,\ (\ :1/2),\ -0.959 + 0.282i)

Particular Values

L(1)L(1) ≈\approx 0.56133712600.5613371260
L(12)L(\frac12) ≈\approx 0.56133712600.5613371260
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1 1
7 1 1
good5 1+0.517T+5T2 1 + 0.517T + 5T^{2}
11 1−1.46T+11T2 1 - 1.46T + 11T^{2}
13 1+(−1.22−2.12i)T+(−6.5+11.2i)T2 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2}
17 1+(1.74+3.01i)T+(−8.5+14.7i)T2 1 + (1.74 + 3.01i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.258−0.448i)T+(−9.5−16.4i)T2 1 + (0.258 - 0.448i)T + (-9.5 - 16.4i)T^{2}
23 1+7.92T+23T2 1 + 7.92T + 23T^{2}
29 1+(−1.36+2.36i)T+(−14.5−25.1i)T2 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2}
31 1+(3.67−6.36i)T+(−15.5−26.8i)T2 1 + (3.67 - 6.36i)T + (-15.5 - 26.8i)T^{2}
37 1+(−4+6.92i)T+(−18.5−32.0i)T2 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2}
41 1+(−2.82−4.89i)T+(−20.5+35.5i)T2 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2}
43 1+(−6.09+10.5i)T+(−21.5−37.2i)T2 1 + (-6.09 + 10.5i)T + (-21.5 - 37.2i)T^{2}
47 1+(2.31+4.00i)T+(−23.5+40.7i)T2 1 + (2.31 + 4.00i)T + (-23.5 + 40.7i)T^{2}
53 1+(−3.36−5.83i)T+(−26.5+45.8i)T2 1 + (-3.36 - 5.83i)T + (-26.5 + 45.8i)T^{2}
59 1+(−7.39+12.8i)T+(−29.5−51.0i)T2 1 + (-7.39 + 12.8i)T + (-29.5 - 51.0i)T^{2}
61 1+(2.19+3.79i)T+(−30.5+52.8i)T2 1 + (2.19 + 3.79i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.90+3.29i)T+(−33.5−58.0i)T2 1 + (-1.90 + 3.29i)T + (-33.5 - 58.0i)T^{2}
71 1−0.803T+71T2 1 - 0.803T + 71T^{2}
73 1+(2.31+4.00i)T+(−36.5+63.2i)T2 1 + (2.31 + 4.00i)T + (-36.5 + 63.2i)T^{2}
79 1+(7.06+12.2i)T+(−39.5+68.4i)T2 1 + (7.06 + 12.2i)T + (-39.5 + 68.4i)T^{2}
83 1+(4.94−8.57i)T+(−41.5−71.8i)T2 1 + (4.94 - 8.57i)T + (-41.5 - 71.8i)T^{2}
89 1+(8.05−13.9i)T+(−44.5−77.0i)T2 1 + (8.05 - 13.9i)T + (-44.5 - 77.0i)T^{2}
97 1+(−0.517+0.896i)T+(−48.5−84.0i)T2 1 + (-0.517 + 0.896i)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

−8.619854584453875773669288528656, −7.888846597774580173378807350242, −7.11654391170977658116426238405, −6.27484518201592522622357314533, −5.35056130533735943602460255666, −4.16435752283130689181889593133, −3.80423722448343225007991847181, −2.51350481575236235270071782156, −1.66450807589180568164678662977, −0.21645570023597463384106240509, 1.27045115915773789580008092107, 2.49315691667108990187887717681, 3.87932181298461306180686263690, 4.35216390635075693218450542485, 5.71152087237810480092399416772, 6.03522775375318443505261177831, 6.96414849926735747411425547893, 7.86789812351547456763733570470, 8.239760211525327233311803555403, 9.107155217165369714781745772483

Graph of the ZZ-function along the critical line