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Properties

Label 2-819-7.2-c1-0-21
Degree $2$
Conductor $819$
Sign $0.975 - 0.220i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.975 - 0.220i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.986960 + 0.110162i\)
\(L(\frac12)\) \(\approx\) \(0.986960 + 0.110162i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.418 + 2.61i)T \)
13 \( 1 - T \)
good2 \( 1 + (0.766 - 1.32i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.266 - 0.460i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.43 + 2.49i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.67 - 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.03 + 1.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.93 + 6.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.04T + 29T^{2} \)
31 \( 1 + (3.11 + 5.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.326 + 0.565i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.59T + 41T^{2} \)
43 \( 1 + 6.10T + 43T^{2} \)
47 \( 1 + (-4.75 + 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.439 - 0.761i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.12 + 1.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.14 + 7.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.19 - 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 + (-0.275 - 0.477i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.80 + 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.87T + 83T^{2} \)
89 \( 1 + (-2.74 + 4.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.27392822332668768233157136515, −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 $Z$-function along the critical line