Properties

Label 2-819-819.139-c0-0-0
Degree $2$
Conductor $819$
Sign $-0.995 - 0.0935i$
Analytic cond. $0.408734$
Root an. cond. $0.639323$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.866 + 0.5i)3-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)6-s + (−0.5 + 0.866i)7-s − 8-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)10-s − 11-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + 0.999·15-s − 16-s + (−0.866 + 0.5i)17-s + (0.499 − 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + 2-s + (−0.866 + 0.5i)3-s + (−0.866 − 0.5i)5-s + (−0.866 + 0.5i)6-s + (−0.5 + 0.866i)7-s − 8-s + (0.499 − 0.866i)9-s + (−0.866 − 0.5i)10-s − 11-s + (−0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + 0.999·15-s − 16-s + (−0.866 + 0.5i)17-s + (0.499 − 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.995 - 0.0935i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ -0.995 - 0.0935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1931764643\)
\(L(\frac12)\) \(\approx\) \(0.1931764643\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 - T + T^{2} \)
5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + T + T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−11.16046666644785936458245698168, −10.01211022351571486964089994832, −9.227683338066229580044824282823, −8.454636224742442380802992411082, −7.15413363094971183052785154253, −6.16471034727461438207079867641, −5.25353406746247396608602035882, −4.75178576327852721300233440380, −3.86344837400627276332761803154, −2.72724094164769865293313061415, 0.15027073046117490147359709343, 2.67815917289982455511262941580, 3.73905880392271394081806112982, 4.69201411489182634797521909661, 5.46386461491144483489677585181, 6.43491238309343556441350916157, 7.44910716223101986076683756530, 7.73394762308907324783502833770, 9.371038350033295547952915098548, 10.33686879352744536556787105350

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