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Properties

Label 2-819-819.412-c0-0-1
Degree $2$
Conductor $819$
Sign $-0.822 + 0.568i$
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  + (−0.5 − 0.866i)2-s − i·3-s + (0.866 − 0.5i)5-s + (−0.866 + 0.5i)6-s + 7-s − 8-s − 9-s + (−0.866 − 0.499i)10-s + (0.5 + 0.866i)11-s − i·13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − i·3-s + (0.866 − 0.5i)5-s + (−0.866 + 0.5i)6-s + 7-s − 8-s − 9-s + (−0.866 − 0.499i)10-s + (0.5 + 0.866i)11-s − i·13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.822 + 0.568i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (412, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ -0.822 + 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9436075410\)
\(L(\frac12)\) \(\approx\) \(0.9436075410\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 - T \)
13 \( 1 + iT \)
good2 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 + 2T + T^{2} \)
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 \( 1 - T^{2} \)
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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

−10.23789551564847444422593784750, −9.145160084580691357198906170668, −8.719552738338096698004922850205, −7.63376422346620424689575609013, −6.67280902556778698565489343659, −5.67979925151830447873884987386, −4.91803273928049618374612338313, −3.06507532298649991736798395559, −1.87072151589929830541150941540, −1.35327513961044349644508019775, 2.23191255081108015681062112832, 3.48466624040421063590918278245, 4.66297605687337423971570491281, 5.78315055244107260913620522363, 6.32935663779118177307284738689, 7.41958254124965460298007860590, 8.358130894526478566837672606986, 9.125058152370480146917099174275, 9.589883867975464073508482241374, 10.71743405253884669359887700806

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