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Properties

Label 2-819-7.4-c1-0-18
Degree $2$
Conductor $819$
Sign $0.165 - 0.986i$
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  + (1.10 + 1.91i)2-s + (−1.44 + 2.51i)4-s + (−1.06 − 1.83i)5-s + (2.63 + 0.272i)7-s − 1.98·8-s + (2.34 − 4.06i)10-s + (2.39 − 4.14i)11-s + 13-s + (2.39 + 5.34i)14-s + (0.697 + 1.20i)16-s + (−1.88 + 3.27i)17-s + (1.78 + 3.08i)19-s + 6.15·20-s + 10.5·22-s + (2.23 + 3.87i)23-s + ⋯
L(s)  = 1  + (0.782 + 1.35i)2-s + (−0.724 + 1.25i)4-s + (−0.474 − 0.822i)5-s + (0.994 + 0.102i)7-s − 0.703·8-s + (0.742 − 1.28i)10-s + (0.721 − 1.25i)11-s + 0.277·13-s + (0.638 + 1.42i)14-s + (0.174 + 0.301i)16-s + (−0.458 + 0.793i)17-s + (0.409 + 0.708i)19-s + 1.37·20-s + 2.25·22-s + (0.466 + 0.807i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.165 - 0.986i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.165 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91928 + 1.62363i\)
\(L(\frac12)\) \(\approx\) \(1.91928 + 1.62363i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.272i)T \)
13 \( 1 - T \)
good2 \( 1 + (-1.10 - 1.91i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.06 + 1.83i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.39 + 4.14i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.88 - 3.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.78 - 3.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.90T + 29T^{2} \)
31 \( 1 + (-1.88 + 3.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.81 + 4.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 3.40T + 43T^{2} \)
47 \( 1 + (3.55 + 6.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.19 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.39 + 4.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.60 + 2.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.44 + 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 + (3.85 - 6.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.58 - 4.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 + (-1.83 - 3.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.40T + 97T^{2} \)
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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

−10.55700577074519834006355907372, −9.037780087432242006869910727277, −8.326160250107991150017411326044, −8.007340163298763068208229264406, −6.82727775055513084595208973815, −5.94089207482769393864247984064, −5.21615834105578046507712550938, −4.33799121873479505115125097298, −3.56522816085926870298947120015, −1.34655831532777280177977784130, 1.34611435391039071731205465820, 2.51562220126975598413937742939, 3.42878302711723011385344953584, 4.64249509231711905919317394734, 4.88139403259339428511881500499, 6.67970964625285003736209104925, 7.28058366306710341990465052609, 8.484220665229477408904868840617, 9.565616410728435341618707511860, 10.42473865585087695764195159678

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