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Properties

Label 2-819-819.139-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.894994539\)
\(L(\frac12)\) \(\approx\) \(1.894994539\)
\(L(1)\) 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

−10.20733059134643817575985011850, −9.595370581318043541114515166098, −8.704754551596408054015355478493, −7.956666869094812225229870364840, −6.67869171330590680229777263096, −5.91080931425162856690422978160, −5.32943966165201254018292418793, −3.75635344926215439432449816326, −2.98383258878554538049267982096, −2.15490333973900978503980154238, 1.99416777801554526391704769188, 3.36935793812062702042592640403, 3.96737384918975649582333947357, 5.03034435387202807877555772342, 5.72716014764884462731516967435, 6.85461172007153829849567392620, 8.031809370844575873288425810843, 8.929106427864171128343135590733, 9.541929591692917409945202423484, 10.36288880318203074012144584332

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