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Properties

Label 2-819-91.37-c0-0-0
Degree $2$
Conductor $819$
Sign $0.101 + 0.994i$
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  − i·4-s + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)13-s − 16-s + (0.5 − 1.86i)19-s + (−0.866 + 0.5i)25-s + (−0.5 + 0.866i)28-s + (0.366 − 1.36i)31-s + (−1.36 + 1.36i)37-s + (1.5 + 0.866i)43-s + (0.499 + 0.866i)49-s + (−0.5 − 0.866i)52-s + (0.866 + 1.5i)61-s + i·64-s + (1.36 − 0.366i)67-s + ⋯
L(s)  = 1  − i·4-s + (−0.866 − 0.5i)7-s + (0.866 − 0.5i)13-s − 16-s + (0.5 − 1.86i)19-s + (−0.866 + 0.5i)25-s + (−0.5 + 0.866i)28-s + (0.366 − 1.36i)31-s + (−1.36 + 1.36i)37-s + (1.5 + 0.866i)43-s + (0.499 + 0.866i)49-s + (−0.5 − 0.866i)52-s + (0.866 + 1.5i)61-s + i·64-s + (1.36 − 0.366i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.101 + 0.994i$
Analytic conductor: \(0.408734\)
Root analytic conductor: \(0.639323\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :0),\ 0.101 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8675582600\)
\(L(\frac12)\) \(\approx\) \(0.8675582600\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
good2 \( 1 + iT^{2} \)
5 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
41 \( 1 + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.5 - 0.133i)T + (0.866 - 0.5i)T^{2} \)
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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

−10.15417480440096369494279407241, −9.542562203174453998820673174653, −8.779760179234158765069740243842, −7.52830314900129629649787823326, −6.64792816955541274882868095414, −5.93872870072312790735649818424, −4.97692369678211545766879074504, −3.84261286866511285283440046434, −2.62014766683597483759708828428, −0.934168193879932023473507735763, 2.08133581470381527109316126890, 3.43697052734072426599579772857, 3.92535616955122374502621529411, 5.46794785908147232081169302640, 6.35194162076480399078872809816, 7.22289073500230645828741298088, 8.206949418454394079114702469997, 8.840478860862747692000131509183, 9.697647144485017313004024398423, 10.61472023428298259404884436914

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