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Properties

Label 2-936-104.101-c1-0-35
Degree $2$
Conductor $936$
Sign $0.383 - 0.923i$
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  + (1.29 + 0.565i)2-s + (1.35 + 1.46i)4-s + 1.25·5-s + (1.81 + 1.05i)7-s + (0.932 + 2.67i)8-s + (1.63 + 0.712i)10-s + (0.586 + 1.01i)11-s + (−2.36 + 2.71i)13-s + (1.76 + 2.39i)14-s + (−0.303 + 3.98i)16-s + (2.24 − 3.89i)17-s + (−0.575 + 0.997i)19-s + (1.71 + 1.84i)20-s + (0.185 + 1.64i)22-s + (−3.69 − 6.40i)23-s + ⋯
L(s)  = 1  + (0.916 + 0.400i)2-s + (0.679 + 0.733i)4-s + 0.563·5-s + (0.687 + 0.397i)7-s + (0.329 + 0.944i)8-s + (0.516 + 0.225i)10-s + (0.176 + 0.306i)11-s + (−0.656 + 0.754i)13-s + (0.471 + 0.638i)14-s + (−0.0757 + 0.997i)16-s + (0.545 − 0.944i)17-s + (−0.132 + 0.228i)19-s + (0.382 + 0.413i)20-s + (0.0395 + 0.351i)22-s + (−0.771 − 1.33i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.383 - 0.923i$
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: \((2,\ 936,\ (\ :1/2),\ 0.383 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.65855 + 1.77556i\)
\(L(\frac12)\) \(\approx\) \(2.65855 + 1.77556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.565i)T \)
3 \( 1 \)
13 \( 1 + (2.36 - 2.71i)T \)
good5 \( 1 - 1.25T + 5T^{2} \)
7 \( 1 + (-1.81 - 1.05i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.586 - 1.01i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.24 + 3.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.575 - 0.997i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.69 + 6.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.34 + 1.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.682iT - 31T^{2} \)
37 \( 1 + (-4.79 - 8.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.84 + 3.95i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.32 - 1.34i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.71iT - 47T^{2} \)
53 \( 1 + 9.95iT - 53T^{2} \)
59 \( 1 + (5.00 - 8.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.93 + 1.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.888 + 1.53i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.05 - 0.609i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.0iT - 73T^{2} \)
79 \( 1 - 2.73T + 79T^{2} \)
83 \( 1 + 3.16T + 83T^{2} \)
89 \( 1 + (4.52 - 2.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.68 + 3.85i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.21313616759118946436538240675, −9.389433565888903828317932656862, −8.319751725584138520086926764380, −7.59121300025005403190969198892, −6.60715553746895412284328284381, −5.87653382295812525315064258023, −4.87671338139395105025378097219, −4.28384066777012866750926670862, −2.76826987602080525966982246854, −1.91235158468991459304485549299, 1.26793529400589888952893568728, 2.39393945263146203256996636881, 3.61276546052918305388684160459, 4.51703165733942634466064985190, 5.61001705357515964970241752457, 6.03142447061502644957703302601, 7.36187604801824282537909812688, 7.990153558275783258444022551553, 9.390074685365379790047898240508, 10.08420650991934056232955031756

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