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Properties

Label 2-936-117.112-c0-0-1
Degree $2$
Conductor $936$
Sign $-0.786 + 0.617i$
Analytic cond. $0.467124$
Root an. cond. $0.683465$
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)3-s + (−1.36 − 0.366i)7-s + (−0.499 − 0.866i)9-s + (−1.36 − 0.366i)11-s + (−0.866 − 0.5i)13-s + i·17-s + (−1 − i)19-s + (1 − 0.999i)21-s + (0.866 + 0.5i)23-s + (−0.866 + 0.5i)25-s + 0.999·27-s + (1 − 0.999i)33-s + (1 − i)37-s + (0.866 − 0.499i)39-s + (−1.36 + 0.366i)41-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−1.36 − 0.366i)7-s + (−0.499 − 0.866i)9-s + (−1.36 − 0.366i)11-s + (−0.866 − 0.5i)13-s + i·17-s + (−1 − i)19-s + (1 − 0.999i)21-s + (0.866 + 0.5i)23-s + (−0.866 + 0.5i)25-s + 0.999·27-s + (1 − 0.999i)33-s + (1 − i)37-s + (0.866 − 0.499i)39-s + (−1.36 + 0.366i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.786 + 0.617i$
Analytic conductor: \(0.467124\)
Root analytic conductor: \(0.683465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :0),\ -0.786 + 0.617i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05462070289\)
\(L(\frac12)\) \(\approx\) \(0.05462070289\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (0.866 - 0.5i)T^{2} \)
7 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + (1 + i)T + iT^{2} \)
23 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 + (1 - i)T - iT^{2} \)
97 \( 1 + (-0.866 - 0.5i)T^{2} \)
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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

−9.958830476349672910991628675185, −9.409530520325551782667901857259, −8.367254649684904328849264828652, −7.30971694400707788089361549531, −6.34053909065931783032001460845, −5.56907634088591951604589899920, −4.66044480531168502521656369356, −3.53894067762455677607740541711, −2.72446859138517158782270158406, −0.05052041601312060385186691248, 2.19091270186842215233725906227, 2.96153207448637234642334022329, 4.61989787995641170815724929487, 5.53972520148029502310579387445, 6.41291468871899552777797448079, 7.10053378785479131063778362179, 7.906933398337334218983767692278, 8.883649100320035699420022098900, 9.991368691020259145861453317115, 10.37783240027285060604298933651

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