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Properties

Label 2-819-91.23-c1-0-1
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: $\chi_{819} (478, \cdot )$
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

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

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