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Properties

Label 2-819-91.5-c1-0-40
Degree $2$
Conductor $819$
Sign $-0.998 - 0.0546i$
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  + (−0.493 − 1.84i)2-s + (−1.41 + 0.818i)4-s + (3.30 − 0.885i)5-s + (−1.12 − 2.39i)7-s + (−0.489 − 0.489i)8-s + (−3.26 − 5.65i)10-s + (0.445 − 1.66i)11-s + (−3.57 − 0.501i)13-s + (−3.84 + 3.26i)14-s + (−2.29 + 3.97i)16-s + (1.22 + 2.12i)17-s + (5.03 − 1.34i)19-s + (−3.96 + 3.96i)20-s − 3.28·22-s + (−3.97 − 2.29i)23-s + ⋯
L(s)  = 1  + (−0.349 − 1.30i)2-s + (−0.708 + 0.409i)4-s + (1.47 − 0.396i)5-s + (−0.426 − 0.904i)7-s + (−0.173 − 0.173i)8-s + (−1.03 − 1.78i)10-s + (0.134 − 0.501i)11-s + (−0.990 − 0.138i)13-s + (−1.02 + 0.871i)14-s + (−0.574 + 0.994i)16-s + (0.297 + 0.515i)17-s + (1.15 − 0.309i)19-s + (−0.885 + 0.885i)20-s − 0.700·22-s + (−0.828 − 0.478i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.998 - 0.0546i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.998 - 0.0546i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0378262 + 1.38231i\)
\(L(\frac12)\) \(\approx\) \(0.0378262 + 1.38231i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 + 2.39i)T \)
13 \( 1 + (3.57 + 0.501i)T \)
good2 \( 1 + (0.493 + 1.84i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-3.30 + 0.885i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.445 + 1.66i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.22 - 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.03 + 1.34i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.97 + 2.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.184T + 29T^{2} \)
31 \( 1 + (-0.659 + 2.46i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.210 - 0.0563i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.63 + 4.63i)T + 41iT^{2} \)
43 \( 1 + 0.562iT - 43T^{2} \)
47 \( 1 + (0.998 + 3.72i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.67 - 4.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-13.9 - 3.73i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.30 + 0.754i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.67 + 1.78i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.70 - 1.70i)T - 71iT^{2} \)
73 \( 1 + (-11.7 - 3.15i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.48 - 2.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.504 + 0.504i)T + 83iT^{2} \)
89 \( 1 + (-1.92 - 7.20i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.0 + 12.0i)T + 97iT^{2} \)
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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

−10.00992728355722000967818121030, −9.428130696776435485351072910785, −8.508961529609518111553763798800, −7.20176402887829489082840052142, −6.21844396723102700971161426221, −5.31731018451949541793925189266, −4.01521668627085393375459697341, −2.91759947572331282105337380236, −1.90040953094337629183178818399, −0.74642864737071556621335178298, 2.04243601120047479428127519342, 3.00618674853783582670732711046, 5.07445938581717638453018162518, 5.57301770328851133713868353701, 6.40115834655779929108820925934, 7.06468456636934511089785813599, 7.963109095067921184965373612883, 9.064391577497576061788886811497, 9.711488367642627722408895982170, 10.01053792106882692114578114657

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