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Properties

Label 2-819-7.2-c1-0-21
Degree 22
Conductor 819819
Sign 0.975−0.220i0.975 - 0.220i
Analytic cond. 6.539746.53974
Root an. cond. 2.557292.55729
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  + (−0.766 + 1.32i)2-s + (−0.173 − 0.300i)4-s + (−0.266 + 0.460i)5-s + (−0.418 − 2.61i)7-s − 2.53·8-s + (−0.407 − 0.705i)10-s + (−1.43 − 2.49i)11-s + 13-s + (3.78 + 1.44i)14-s + (2.28 − 3.96i)16-s + (1.67 + 2.89i)17-s + (1.03 − 1.78i)19-s + 0.184·20-s + 4.41·22-s + (3.93 − 6.81i)23-s + ⋯
L(s)  = 1  + (−0.541 + 0.938i)2-s + (−0.0868 − 0.150i)4-s + (−0.118 + 0.206i)5-s + (−0.158 − 0.987i)7-s − 0.895·8-s + (−0.128 − 0.223i)10-s + (−0.434 − 0.751i)11-s + 0.277·13-s + (1.01 + 0.386i)14-s + (0.571 − 0.990i)16-s + (0.405 + 0.703i)17-s + (0.236 − 0.410i)19-s + 0.0413·20-s + 0.940·22-s + (0.819 − 1.42i)23-s + ⋯

Functional equation

Λ(s)=(819s/2 ΓC(s) L(s)=((0.975−0.220i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(819s/2 ΓC(s+1/2) L(s)=((0.975−0.220i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 819819    =    32⋅7⋅133^{2} \cdot 7 \cdot 13
Sign: 0.975−0.220i0.975 - 0.220i
Analytic conductor: 6.539746.53974
Root analytic conductor: 2.557292.55729
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ819(352,⋅)\chi_{819} (352, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 819, ( :1/2), 0.975−0.220i)(2,\ 819,\ (\ :1/2),\ 0.975 - 0.220i)

Particular Values

L(1)L(1) ≈\approx 0.986960+0.110162i0.986960 + 0.110162i
L(12)L(\frac12) ≈\approx 0.986960+0.110162i0.986960 + 0.110162i
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)
bad3 1 1
7 1+(0.418+2.61i)T 1 + (0.418 + 2.61i)T
13 1−T 1 - T
good2 1+(0.766−1.32i)T+(−1−1.73i)T2 1 + (0.766 - 1.32i)T + (-1 - 1.73i)T^{2}
5 1+(0.266−0.460i)T+(−2.5−4.33i)T2 1 + (0.266 - 0.460i)T + (-2.5 - 4.33i)T^{2}
11 1+(1.43+2.49i)T+(−5.5+9.52i)T2 1 + (1.43 + 2.49i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1.67−2.89i)T+(−8.5+14.7i)T2 1 + (-1.67 - 2.89i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.03+1.78i)T+(−9.5−16.4i)T2 1 + (-1.03 + 1.78i)T + (-9.5 - 16.4i)T^{2}
23 1+(−3.93+6.81i)T+(−11.5−19.9i)T2 1 + (-3.93 + 6.81i)T + (-11.5 - 19.9i)T^{2}
29 1−7.04T+29T2 1 - 7.04T + 29T^{2}
31 1+(3.11+5.39i)T+(−15.5+26.8i)T2 1 + (3.11 + 5.39i)T + (-15.5 + 26.8i)T^{2}
37 1+(−0.326+0.565i)T+(−18.5−32.0i)T2 1 + (-0.326 + 0.565i)T + (-18.5 - 32.0i)T^{2}
41 1−4.59T+41T2 1 - 4.59T + 41T^{2}
43 1+6.10T+43T2 1 + 6.10T + 43T^{2}
47 1+(−4.75+8.24i)T+(−23.5−40.7i)T2 1 + (-4.75 + 8.24i)T + (-23.5 - 40.7i)T^{2}
53 1+(−0.439−0.761i)T+(−26.5+45.8i)T2 1 + (-0.439 - 0.761i)T + (-26.5 + 45.8i)T^{2}
59 1+(1.12+1.94i)T+(−29.5+51.0i)T2 1 + (1.12 + 1.94i)T + (-29.5 + 51.0i)T^{2}
61 1+(−4.14+7.18i)T+(−30.5−52.8i)T2 1 + (-4.14 + 7.18i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.19−10.7i)T+(−33.5+58.0i)T2 1 + (-6.19 - 10.7i)T + (-33.5 + 58.0i)T^{2}
71 1+10.6T+71T2 1 + 10.6T + 71T^{2}
73 1+(−0.275−0.477i)T+(−36.5+63.2i)T2 1 + (-0.275 - 0.477i)T + (-36.5 + 63.2i)T^{2}
79 1+(−5.80+10.0i)T+(−39.5−68.4i)T2 1 + (-5.80 + 10.0i)T + (-39.5 - 68.4i)T^{2}
83 1−5.87T+83T2 1 - 5.87T + 83T^{2}
89 1+(−2.74+4.75i)T+(−44.5−77.0i)T2 1 + (-2.74 + 4.75i)T + (-44.5 - 77.0i)T^{2}
97 1−12.7T+97T2 1 - 12.7T + 97T^{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

−10.27392822332668768233157136515, −9.145446205975690924680151245337, −8.403036890930933737498287529830, −7.69429630837381503081032971521, −6.87886057288019245763374633907, −6.24919907504026180994108254633, −5.14362230866252066859423335310, −3.79348466323644940347311047065, −2.84645808086428443145064051981, −0.67070803828216033536479638577, 1.23433213916247244576929954729, 2.48886052745411542826786170203, 3.30152091762462332257994884821, 4.88267001575096858109643153482, 5.69974743546478878122359763878, 6.76055451100095370443383071154, 7.900573193725361109694129736915, 8.848909399605993751709541388924, 9.445053599299110631466046053416, 10.16946165200984479052039629130

Graph of the ZZ-function along the critical line