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Properties

Label 2-819-91.9-c1-0-24
Degree $2$
Conductor $819$
Sign $0.264 - 0.964i$
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  + (1.14 + 1.97i)2-s + (−1.59 + 2.77i)4-s + (1.46 − 2.54i)5-s + (−2.34 − 1.23i)7-s − 2.73·8-s + 6.69·10-s + 5.16·11-s + (−0.364 + 3.58i)13-s + (−0.233 − 6.02i)14-s + (0.0801 + 0.138i)16-s + (2.52 − 4.37i)17-s + 2.25·19-s + (4.69 + 8.13i)20-s + (5.89 + 10.2i)22-s + (2.61 + 4.53i)23-s + ⋯
L(s)  = 1  + (0.806 + 1.39i)2-s + (−0.799 + 1.38i)4-s + (0.656 − 1.13i)5-s + (−0.884 − 0.466i)7-s − 0.967·8-s + 2.11·10-s + 1.55·11-s + (−0.101 + 0.994i)13-s + (−0.0623 − 1.61i)14-s + (0.0200 + 0.0346i)16-s + (0.612 − 1.06i)17-s + 0.518·19-s + (1.05 + 1.82i)20-s + (1.25 + 2.17i)22-s + (0.545 + 0.944i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.264 - 0.964i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.264 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07671 + 1.58418i\)
\(L(\frac12)\) \(\approx\) \(2.07671 + 1.58418i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.34 + 1.23i)T \)
13 \( 1 + (0.364 - 3.58i)T \)
good2 \( 1 + (-1.14 - 1.97i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-1.46 + 2.54i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 5.16T + 11T^{2} \)
17 \( 1 + (-2.52 + 4.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 2.25T + 19T^{2} \)
23 \( 1 + (-2.61 - 4.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.216 - 0.375i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.34 + 2.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.12 - 3.67i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.269 - 0.466i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.66 + 8.07i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.87 + 8.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.377 + 0.653i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.82 - 3.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.95T + 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 + (3.90 + 6.76i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.94 + 13.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.79 - 13.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + (2.00 + 3.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.69 - 4.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.00134995790606913050211063051, −9.230995534220021756011933757432, −8.830633100157833227348341768555, −7.39335940047408901237555983052, −6.89383841485443661822631901231, −6.01008004218557172954798453464, −5.23830624804957385143718995140, −4.35049926434681867989076901498, −3.48480099137728640825194119811, −1.33512573575542749221225344113, 1.39229457240910654099614710268, 2.74995431902146623753182029755, 3.23246299103238728813617630152, 4.23858820340699817306244084828, 5.68733889933060569944699889510, 6.22018066731944667592181188132, 7.22675823720693515439130625044, 8.751895892604816773898936139279, 9.756389834491036529549584378328, 10.16251285631956659399898765708

Graph of the $Z$-function along the critical line