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Properties

Label 2-819-273.191-c0-0-0
Degree $2$
Conductor $819$
Sign $0.292 - 0.956i$
Analytic cond. $0.408734$
Root an. cond. $0.639323$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.292 - 0.956i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.292 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5678384715\)
\(L(\frac12)\) \(\approx\) \(0.5678384715\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line