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Properties

Label 2-936-104.99-c1-0-7
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: $\chi_{936} (307, \cdot )$
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

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

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