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Properties

Label 2-819-13.9-c1-0-2
Degree $2$
Conductor $819$
Sign $-0.500 + 0.866i$
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  + (−0.636 + 1.10i)2-s + (0.188 + 0.326i)4-s − 1.10·5-s + (0.5 + 0.866i)7-s − 3.02·8-s + (0.702 − 1.21i)10-s + (−0.174 + 0.302i)11-s + (−3.47 + 0.955i)13-s − 1.27·14-s + (1.55 − 2.68i)16-s + (0.363 + 0.628i)17-s + (−1.15 − 1.99i)19-s + (−0.208 − 0.360i)20-s + (−0.222 − 0.384i)22-s + (0.0375 − 0.0649i)23-s + ⋯
L(s)  = 1  + (−0.450 + 0.780i)2-s + (0.0943 + 0.163i)4-s − 0.493·5-s + (0.188 + 0.327i)7-s − 1.07·8-s + (0.222 − 0.384i)10-s + (−0.0525 + 0.0911i)11-s + (−0.964 + 0.265i)13-s − 0.340·14-s + (0.387 − 0.671i)16-s + (0.0880 + 0.152i)17-s + (−0.264 − 0.457i)19-s + (−0.0465 − 0.0805i)20-s + (−0.0473 − 0.0820i)22-s + (0.00782 − 0.0135i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.500 + 0.866i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (568, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.500 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107626 - 0.186413i\)
\(L(\frac12)\) \(\approx\) \(0.107626 - 0.186413i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (3.47 - 0.955i)T \)
good2 \( 1 + (0.636 - 1.10i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 1.10T + 5T^{2} \)
11 \( 1 + (0.174 - 0.302i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.363 - 0.628i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.15 + 1.99i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0375 + 0.0649i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.240 + 0.416i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.85T + 31T^{2} \)
37 \( 1 + (-0.551 + 0.955i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.54 + 2.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.31 - 4.00i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.70T + 47T^{2} \)
53 \( 1 + 5.54T + 53T^{2} \)
59 \( 1 + (1.96 + 3.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.77 + 4.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.06 - 7.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.76 + 8.25i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 17.1T + 79T^{2} \)
83 \( 1 - 11.9T + 83T^{2} \)
89 \( 1 + (5.98 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.05 + 8.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.83074053836473394653183973822, −9.594815646722643729762469930291, −9.031067773818660393410319602723, −8.019551525649661008716514282291, −7.53161122211895696997628763783, −6.68093359784498937768641085333, −5.73548525832770753350347106308, −4.64863124179717388786171034863, −3.43584852617104891801103962094, −2.22631395358086750511450063698, 0.11570016734611527429738049951, 1.66829480928252681761851912434, 2.85661752914604382586139877206, 3.94446422800072620836887025855, 5.15914489142836206311254408116, 6.14457820894860499631635039537, 7.27550513398037485493065900667, 8.013277340531777632471635255693, 9.070856001819397951722994541243, 9.811143513396122112360211371535

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