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Properties

Label 2-819-819.328-c0-0-1
Degree $2$
Conductor $819$
Sign $0.822 + 0.568i$
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  + (−0.5 + 0.866i)2-s − i·3-s + (−0.866 − 0.5i)5-s + (0.866 + 0.5i)6-s + 7-s − 8-s − 9-s + (0.866 − 0.499i)10-s + (0.5 − 0.866i)11-s − i·13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)18-s + (−0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s − i·3-s + (−0.866 − 0.5i)5-s + (0.866 + 0.5i)6-s + 7-s − 8-s − 9-s + (0.866 − 0.499i)10-s + (0.5 − 0.866i)11-s − i·13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)18-s + (−0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.822 + 0.568i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.822 + 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6878120750\)
\(L(\frac12)\) \(\approx\) \(0.6878120750\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.41802611935113074274986304578, −8.741502766749976776856975267265, −8.473539114998791054319372722749, −7.87965118591515241636831389606, −7.20580455426322755028016578321, −6.14889978536158819979786214351, −5.43741690821658506793918330184, −3.98389600702855193801712875269, −2.70609444675800472559588073046, −0.889396325692088119020345675754, 1.73281158551708641052381449247, 3.02112894217861435190449591608, 4.06962722804624097351802639807, 4.83408571729965777185374827687, 6.11658136240227932038631602888, 7.23987494747989501123056617395, 8.295011829656140946017713180545, 9.051853996844306583921864469087, 9.884899331427009290046113166777, 10.53854399045989399516714828311

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