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Properties

Label 2-936-104.101-c1-0-4
Degree $2$
Conductor $936$
Sign $-0.883 + 0.468i$
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  + (0.0207 + 1.41i)2-s + (−1.99 + 0.0586i)4-s + 0.642·5-s + (0.306 + 0.176i)7-s + (−0.124 − 2.82i)8-s + (0.0133 + 0.908i)10-s + (−1.15 − 2.00i)11-s + (−3.23 + 1.58i)13-s + (−0.243 + 0.437i)14-s + (3.99 − 0.234i)16-s + (−3.34 + 5.79i)17-s + (−1.51 + 2.61i)19-s + (−1.28 + 0.0376i)20-s + (2.80 − 1.67i)22-s + (−0.693 − 1.20i)23-s + ⋯
L(s)  = 1  + (0.0146 + 0.999i)2-s + (−0.999 + 0.0293i)4-s + 0.287·5-s + (0.115 + 0.0668i)7-s + (−0.0439 − 0.999i)8-s + (0.00421 + 0.287i)10-s + (−0.348 − 0.604i)11-s + (−0.897 + 0.440i)13-s + (−0.0651 + 0.116i)14-s + (0.998 − 0.0586i)16-s + (−0.811 + 1.40i)17-s + (−0.346 + 0.600i)19-s + (−0.287 + 0.00842i)20-s + (0.598 − 0.357i)22-s + (−0.144 − 0.250i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.883 + 0.468i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.883 + 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127201 - 0.511668i\)
\(L(\frac12)\) \(\approx\) \(0.127201 - 0.511668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0207 - 1.41i)T \)
3 \( 1 \)
13 \( 1 + (3.23 - 1.58i)T \)
good5 \( 1 - 0.642T + 5T^{2} \)
7 \( 1 + (-0.306 - 0.176i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.15 + 2.00i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.34 - 5.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.51 - 2.61i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.693 + 1.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.85 - 3.38i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.83iT - 31T^{2} \)
37 \( 1 + (2.54 + 4.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.03 + 4.05i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0281 + 0.0162i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.6iT - 47T^{2} \)
53 \( 1 + 3.40iT - 53T^{2} \)
59 \( 1 + (-2.40 + 4.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.23 + 4.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.24 + 3.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.35 - 3.08i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.24iT - 73T^{2} \)
79 \( 1 - 7.68T + 79T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 + (-1.37 + 0.796i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.29 + 3.05i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.39409897281490972064131764555, −9.500433636331111904936055037100, −8.725731015539409838369779539746, −8.016244839111766479783758790923, −7.12997534061687465014011609075, −6.21465498232980937976549023156, −5.54525427047971670095433438864, −4.52117667276213599805821115523, −3.56583459405433226759657440238, −1.90077047471860625480955833384, 0.23237334533815564607567195574, 2.05787523259618107341304278698, 2.76840403261151350969440060835, 4.19428054925455558582106407021, 4.92012929956867339075040478708, 5.84411093601873877178892878052, 7.25083235739049288039879431721, 7.930914864717169214651052636967, 9.164732777888366312499499713066, 9.635377194011944583510147721899

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