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Properties

Label 2-936-104.83-c1-0-36
Degree $2$
Conductor $936$
Sign $-0.367 + 0.929i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 − 0.189i)2-s + (1.92 + 0.531i)4-s + (−0.612 − 0.612i)5-s + (−1.94 + 1.94i)7-s + (−2.60 − 1.10i)8-s + (0.741 + 0.973i)10-s + (−0.148 − 0.148i)11-s + (−0.952 + 3.47i)13-s + (3.09 − 2.36i)14-s + (3.43 + 2.04i)16-s − 3.60i·17-s + (1.89 − 1.89i)19-s + (−0.855 − 1.50i)20-s + (0.180 + 0.236i)22-s − 2.04·23-s + ⋯
L(s)  = 1  + (−0.990 − 0.133i)2-s + (0.964 + 0.265i)4-s + (−0.273 − 0.273i)5-s + (−0.736 + 0.736i)7-s + (−0.919 − 0.392i)8-s + (0.234 + 0.307i)10-s + (−0.0448 − 0.0448i)11-s + (−0.264 + 0.964i)13-s + (0.828 − 0.630i)14-s + (0.859 + 0.511i)16-s − 0.875i·17-s + (0.434 − 0.434i)19-s + (−0.191 − 0.336i)20-s + (0.0384 + 0.0504i)22-s − 0.426·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.367 + 0.929i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.367 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.257105 - 0.378116i\)
\(L(\frac12)\) \(\approx\) \(0.257105 - 0.378116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.189i)T \)
3 \( 1 \)
13 \( 1 + (0.952 - 3.47i)T \)
good5 \( 1 + (0.612 + 0.612i)T + 5iT^{2} \)
7 \( 1 + (1.94 - 1.94i)T - 7iT^{2} \)
11 \( 1 + (0.148 + 0.148i)T + 11iT^{2} \)
17 \( 1 + 3.60iT - 17T^{2} \)
19 \( 1 + (-1.89 + 1.89i)T - 19iT^{2} \)
23 \( 1 + 2.04T + 23T^{2} \)
29 \( 1 + 6.31iT - 29T^{2} \)
31 \( 1 + (0.261 + 0.261i)T + 31iT^{2} \)
37 \( 1 + (-1.73 + 1.73i)T - 37iT^{2} \)
41 \( 1 + (-0.454 + 0.454i)T - 41iT^{2} \)
43 \( 1 + 10.8iT - 43T^{2} \)
47 \( 1 + (6.24 - 6.24i)T - 47iT^{2} \)
53 \( 1 + 7.10iT - 53T^{2} \)
59 \( 1 + (-7.60 - 7.60i)T + 59iT^{2} \)
61 \( 1 + 5.54iT - 61T^{2} \)
67 \( 1 + (1.25 - 1.25i)T - 67iT^{2} \)
71 \( 1 + (7.84 + 7.84i)T + 71iT^{2} \)
73 \( 1 + (5.73 + 5.73i)T + 73iT^{2} \)
79 \( 1 + 6.43iT - 79T^{2} \)
83 \( 1 + (3.01 - 3.01i)T - 83iT^{2} \)
89 \( 1 + (7.13 + 7.13i)T + 89iT^{2} \)
97 \( 1 + (-7.21 + 7.21i)T - 97iT^{2} \)
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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

−9.575334035724820542634863343086, −9.136090076408772302657102376104, −8.289741244084157875376077330139, −7.36079803106387233823207683624, −6.56665384111137601964004173410, −5.69628866886378531908299293576, −4.34739269918799659238483440960, −3.03341034932129985234842519642, −2.09843690237454131288852007799, −0.31051396270194781612017455375, 1.28608690575436385126669132507, 2.90547808147104914940738152523, 3.73334517377089723044361172628, 5.34100438776085642297747883726, 6.29761373398339613817993952408, 7.12328956458739813136260805896, 7.79492898792436912546401658919, 8.571461917279975069375121247889, 9.678909406035871901555331145263, 10.17894432760998448419545729098

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