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Properties

Label 2-819-7.4-c1-0-27
Degree $2$
Conductor $819$
Sign $-0.988 - 0.152i$
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  + (−1.00 − 1.73i)2-s + (−1.01 + 1.76i)4-s + (0.452 + 0.784i)5-s + (0.237 − 2.63i)7-s + 0.0686·8-s + (0.909 − 1.57i)10-s + (0.358 − 0.620i)11-s + 13-s + (−4.82 + 2.23i)14-s + (1.96 + 3.40i)16-s + (1.17 − 2.03i)17-s + (−3.31 − 5.74i)19-s − 1.84·20-s − 1.43·22-s + (1.87 + 3.25i)23-s + ⋯
L(s)  = 1  + (−0.710 − 1.22i)2-s + (−0.508 + 0.880i)4-s + (0.202 + 0.350i)5-s + (0.0898 − 0.995i)7-s + 0.0242·8-s + (0.287 − 0.498i)10-s + (0.107 − 0.187i)11-s + 0.277·13-s + (−1.28 + 0.596i)14-s + (0.491 + 0.850i)16-s + (0.285 − 0.494i)17-s + (−0.761 − 1.31i)19-s − 0.411·20-s − 0.306·22-s + (0.391 + 0.678i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.988 - 0.152i$
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: \((2,\ 819,\ (\ :1/2),\ -0.988 - 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0596455 + 0.776416i\)
\(L(\frac12)\) \(\approx\) \(0.0596455 + 0.776416i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.237 + 2.63i)T \)
13 \( 1 - T \)
good2 \( 1 + (1.00 + 1.73i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.452 - 0.784i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.358 + 0.620i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.17 + 2.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.31 + 5.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.87 - 3.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.25T + 29T^{2} \)
31 \( 1 + (0.785 - 1.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.60 + 4.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.92T + 41T^{2} \)
43 \( 1 + 9.43T + 43T^{2} \)
47 \( 1 + (4.15 + 7.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.04 + 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.358 + 0.620i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.82 - 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.69 - 8.13i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + (-1.73 + 3.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.50 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.54T + 83T^{2} \)
89 \( 1 + (-6.02 - 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.43T + 97T^{2} \)
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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

−10.19957110445087743368757547627, −9.080484685108106010838407337708, −8.479894348477344786500269995590, −7.25695036013544703352802814635, −6.53541897372156821943596604988, −5.16002000159369071949066907910, −3.88891176430757099177509461497, −3.01833097835413432971855636222, −1.82181798778943455566236309451, −0.48900628888699065437638740318, 1.67507372826531123125784908377, 3.27296697689257068530349028666, 4.81003002193525833056082929841, 5.76739916057422830271231563315, 6.29863464227683865810620657305, 7.31883538255412257868641724415, 8.357164912130622511174960618313, 8.637209127535078916010709806389, 9.539381854492194714103816848278, 10.30922882051567580304372289271

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