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Properties

Label 2-819-91.4-c1-0-44
Degree $2$
Conductor $819$
Sign $0.330 - 0.943i$
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  − 2.30i·2-s − 3.30·4-s + (−0.733 + 0.423i)5-s + (−0.357 − 2.62i)7-s + 3.00i·8-s + (0.975 + 1.69i)10-s + (−1.30 + 0.751i)11-s + (−2.92 + 2.11i)13-s + (−6.03 + 0.824i)14-s + 0.313·16-s + 2.07·17-s + (0.0410 + 0.0237i)19-s + (2.42 − 1.40i)20-s + (1.73 + 2.99i)22-s − 7.81·23-s + ⋯
L(s)  = 1  − 1.62i·2-s − 1.65·4-s + (−0.328 + 0.189i)5-s + (−0.135 − 0.990i)7-s + 1.06i·8-s + (0.308 + 0.534i)10-s + (−0.392 + 0.226i)11-s + (−0.810 + 0.585i)13-s + (−1.61 + 0.220i)14-s + 0.0782·16-s + 0.502·17-s + (0.00942 + 0.00544i)19-s + (0.542 − 0.313i)20-s + (0.369 + 0.639i)22-s − 1.63·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.330 - 0.943i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.330 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.158570 + 0.112489i\)
\(L(\frac12)\) \(\approx\) \(0.158570 + 0.112489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.357 + 2.62i)T \)
13 \( 1 + (2.92 - 2.11i)T \)
good2 \( 1 + 2.30iT - 2T^{2} \)
5 \( 1 + (0.733 - 0.423i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.30 - 0.751i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.07T + 17T^{2} \)
19 \( 1 + (-0.0410 - 0.0237i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.81T + 23T^{2} \)
29 \( 1 + (-0.679 + 1.17i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.80 + 3.93i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.70iT - 37T^{2} \)
41 \( 1 + (-8.67 - 5.00i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.63 - 8.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.311 + 0.180i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.35 + 2.34i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.64iT - 59T^{2} \)
61 \( 1 + (2.26 - 3.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.76 - 1.02i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.3 - 7.10i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.85 - 3.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.82 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 + 17.5iT - 89T^{2} \)
97 \( 1 + (-0.369 + 0.213i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.825833396177188890636937446375, −9.220207426068496631117609629600, −7.80455757961383449354054718668, −7.26980995525451138397701934014, −5.83164738855821419334687970288, −4.41023045512273212324783586683, −3.94271627699745681588164077677, −2.80057601968878910493028596866, −1.67730944594870939693081538607, −0.092538717615619190739759191759, 2.46840036802076855308653574056, 3.97535707349629238169125239524, 5.19146037103840292130380615322, 5.65879920096198472177271167374, 6.54902174758053979974448879529, 7.65607827300515465371088198247, 8.067623956610351725023003262113, 8.922832795288235647825790124847, 9.711312595422040167578232979235, 10.72904083501557514498404591766

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