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Properties

Label 2-819-273.191-c0-0-0
Degree 22
Conductor 819819
Sign 0.292−0.956i0.292 - 0.956i
Analytic cond. 0.4087340.408734
Root an. cond. 0.6393230.639323
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 0.707i)2-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)13-s + (−1.22 − 0.707i)14-s + (0.499 + 0.866i)16-s + (1.22 + 0.707i)17-s − 19-s + (−0.5 − 0.866i)25-s + 1.41i·26-s + 0.999·28-s + (1.22 + 0.707i)29-s + (−1.22 − 0.707i)32-s − 2·34-s + (0.5 + 0.866i)37-s + (1.22 − 0.707i)38-s + ⋯
L(s)  = 1  + (−1.22 + 0.707i)2-s + (0.499 − 0.866i)4-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)13-s + (−1.22 − 0.707i)14-s + (0.499 + 0.866i)16-s + (1.22 + 0.707i)17-s − 19-s + (−0.5 − 0.866i)25-s + 1.41i·26-s + 0.999·28-s + (1.22 + 0.707i)29-s + (−1.22 − 0.707i)32-s − 2·34-s + (0.5 + 0.866i)37-s + (1.22 − 0.707i)38-s + ⋯

Functional equation

Λ(s)=(819s/2 ΓC(s) L(s)=((0.292−0.956i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(819s/2 ΓC(s) L(s)=((0.292−0.956i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 819819    =    32⋅7⋅133^{2} \cdot 7 \cdot 13
Sign: 0.292−0.956i0.292 - 0.956i
Analytic conductor: 0.4087340.408734
Root analytic conductor: 0.6393230.639323
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ819(737,⋅)\chi_{819} (737, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 819, ( :0), 0.292−0.956i)(2,\ 819,\ (\ :0),\ 0.292 - 0.956i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.56783847150.5678384715
L(12)L(\frac12) ≈\approx 0.56783847150.5678384715
L(1)L(1) 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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
13 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good2 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1−T2 1 - T^{2}
17 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
19 1+T+T2 1 + T + T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(1.22+0.707i)T+(0.5+0.866i)T2 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2}
61 1+T+T2 1 + T + T^{2}
67 1+T2 1 + T^{2}
71 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
73 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)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

−10.36247024347703626352060084349, −9.659282789827569869731818830895, −8.556562843152410987362575159360, −8.302974289659342366863397177415, −7.51745066233653330189657502677, −6.25042921208385036725756078879, −5.81231028199544160500070765772, −4.42832239049009941130756945918, −2.97311759380696937168707192460, −1.37575471875064336708491984616, 1.06917634820741495441891704281, 2.24123307679474711885116250259, 3.64338929166389499606101001987, 4.73628458157619094341009409536, 6.00101326702660964681364144657, 7.28343010092336858430678635223, 7.83077786264992390561945847343, 8.798200116192005834031617153845, 9.435499615580445384153977048047, 10.38095783077850320202194470846

Graph of the ZZ-function along the critical line