Properties

Label 2-819-91.80-c1-0-2
Degree $2$
Conductor $819$
Sign $0.638 + 0.769i$
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  + (−0.680 − 2.53i)2-s + (−4.25 + 2.45i)4-s + (−0.134 + 0.502i)5-s + (−2.56 − 0.650i)7-s + (5.41 + 5.41i)8-s + 1.36·10-s + (−1.41 − 1.41i)11-s + (−2.40 + 2.68i)13-s + (0.0921 + 6.95i)14-s + (5.15 − 8.92i)16-s + (−2.54 − 4.41i)17-s + (5.09 + 5.09i)19-s + (−0.662 − 2.47i)20-s + (−2.62 + 4.55i)22-s + (4.14 + 2.39i)23-s + ⋯
L(s)  = 1  + (−0.481 − 1.79i)2-s + (−2.12 + 1.22i)4-s + (−0.0602 + 0.224i)5-s + (−0.969 − 0.245i)7-s + (1.91 + 1.91i)8-s + 0.432·10-s + (−0.426 − 0.426i)11-s + (−0.667 + 0.744i)13-s + (0.0246 + 1.85i)14-s + (1.28 − 2.23i)16-s + (−0.617 − 1.06i)17-s + (1.16 + 1.16i)19-s + (−0.148 − 0.552i)20-s + (−0.560 + 0.970i)22-s + (0.864 + 0.499i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617826 - 0.290247i\)
\(L(\frac12)\) \(\approx\) \(0.617826 - 0.290247i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.56 + 0.650i)T \)
13 \( 1 + (2.40 - 2.68i)T \)
good2 \( 1 + (0.680 + 2.53i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.134 - 0.502i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.41 + 1.41i)T + 11iT^{2} \)
17 \( 1 + (2.54 + 4.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.09 - 5.09i)T + 19iT^{2} \)
23 \( 1 + (-4.14 - 2.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.35 - 2.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.23 + 0.867i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-7.17 + 1.92i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.938 + 3.50i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.62 + 2.09i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.0803 + 0.0215i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.85 - 11.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.34 - 1.43i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 0.753iT - 61T^{2} \)
67 \( 1 + (7.68 - 7.68i)T - 67iT^{2} \)
71 \( 1 + (-1.33 - 5.00i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.551 - 2.05i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.17 - 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.09 - 7.09i)T + 83iT^{2} \)
89 \( 1 + (3.53 + 13.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.28 + 0.880i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.14660618462434763453165697411, −9.460695560671072793599888742898, −8.994196696254421826704145954452, −7.75421536550183254511411331458, −6.90208814757609787910198551664, −5.36213342165401529799595235962, −4.26516206194523395249158231842, −3.20544638275888248083548374453, −2.61701931826443254641106780302, −1.02756700164547711480737666619, 0.51212377398575184939506748068, 2.92439349132851853163723756774, 4.60330007533316795553675767598, 5.18247914872025655235557385576, 6.31696219051372486524313405745, 6.80377877304163994335031450731, 7.77111690654669425000774557004, 8.452185827182102863129365018155, 9.361866152107552456391419218833, 9.857721007210558950807869187858

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