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Properties

Label 2-936-117.70-c0-0-1
Degree $2$
Conductor $936$
Sign $0.617 - 0.786i$
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)5-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)13-s + (0.366 + 1.36i)15-s − i·17-s + (−0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s − 0.999·27-s + (−1.36 − 0.366i)31-s + (−1 − i)37-s + (0.499 − 0.866i)39-s + (1.36 + 0.366i)41-s + (−0.866 − 0.5i)43-s + (−0.999 + 0.999i)45-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (1.36 + 0.366i)5-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)13-s + (0.366 + 1.36i)15-s − i·17-s + (−0.866 + 0.5i)23-s + (0.866 + 0.5i)25-s − 0.999·27-s + (−1.36 − 0.366i)31-s + (−1 − i)37-s + (0.499 − 0.866i)39-s + (1.36 + 0.366i)41-s + (−0.866 − 0.5i)43-s + (−0.999 + 0.999i)45-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(0.467124\)
Root analytic conductor: \(0.683465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :0),\ 0.617 - 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.374015674\)
\(L(\frac12)\) \(\approx\) \(1.374015674\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
7 \( 1 + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-1 - i)T + iT^{2} \)
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 \( 1 + iT^{2} \)
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

−10.32751613516854217594608569706, −9.488517803539069044719904894606, −9.122868575830957999130430945898, −7.905647972012814351218688878004, −7.07459483921037361991889090942, −5.66711354237621127356006277412, −5.41008212271213825754309856141, −4.07275626799507185591296382738, −2.90562487147164154701597476671, −2.10133350906273524177809593040, 1.64386826607354710263638273602, 2.25241390129955445519990050153, 3.68168941404254976123912345850, 5.03212861946125897489745748922, 6.06114092603797961336957786450, 6.59392283560290303042138392316, 7.63234098919755376672237916712, 8.583305114507488194226785525560, 9.239739349638760949831844354883, 9.922821375376680375672689662101

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