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Properties

Label 2-936-104.29-c1-0-12
Degree $2$
Conductor $936$
Sign $-0.890 + 0.454i$
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  + (0.782 + 1.17i)2-s + (−0.774 + 1.84i)4-s + 2.59i·5-s + (−0.300 + 0.520i)7-s + (−2.77 + 0.530i)8-s + (−3.06 + 2.03i)10-s + (−2.40 + 1.39i)11-s + (−3.60 + 0.0531i)13-s + (−0.848 + 0.0534i)14-s + (−2.79 − 2.85i)16-s + (1.29 − 2.24i)17-s + (4.38 + 2.52i)19-s + (−4.79 − 2.01i)20-s + (−3.52 − 1.74i)22-s + (−2.94 − 5.09i)23-s + ⋯
L(s)  = 1  + (0.553 + 0.832i)2-s + (−0.387 + 0.921i)4-s + 1.16i·5-s + (−0.113 + 0.196i)7-s + (−0.982 + 0.187i)8-s + (−0.968 + 0.643i)10-s + (−0.726 + 0.419i)11-s + (−0.999 + 0.0147i)13-s + (−0.226 + 0.0142i)14-s + (−0.699 − 0.714i)16-s + (0.313 − 0.543i)17-s + (1.00 + 0.580i)19-s + (−1.07 − 0.450i)20-s + (−0.751 − 0.372i)22-s + (−0.613 − 1.06i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.890 + 0.454i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.890 + 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290548 - 1.20908i\)
\(L(\frac12)\) \(\approx\) \(0.290548 - 1.20908i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.782 - 1.17i)T \)
3 \( 1 \)
13 \( 1 + (3.60 - 0.0531i)T \)
good5 \( 1 - 2.59iT - 5T^{2} \)
7 \( 1 + (0.300 - 0.520i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.40 - 1.39i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.29 + 2.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.38 - 2.52i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.94 + 5.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.54 - 0.889i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.09T + 31T^{2} \)
37 \( 1 + (8.82 - 5.09i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.26 - 9.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.44 + 4.87i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.13T + 47T^{2} \)
53 \( 1 - 6.01iT - 53T^{2} \)
59 \( 1 + (-9.44 - 5.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.84 - 1.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.39 + 1.95i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.230 - 0.399i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 + 7.83T + 79T^{2} \)
83 \( 1 - 0.930iT - 83T^{2} \)
89 \( 1 + (1.17 + 2.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.91 + 10.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.26803194584505683831571378188, −9.881642207076944185157306269263, −8.632860761787229457927405117170, −7.66839313003302912302795823845, −7.15938034733210075887983257859, −6.34475604688626121805613815019, −5.37632281300394036367763205679, −4.54143951662233273071127086208, −3.20070655272286313890199062107, −2.56134626206803386765210035627, 0.46668969336645259835736756221, 1.83741795979186527437394573727, 3.13086849993547534174687956982, 4.15960282582045478209977033907, 5.26595825136387408552474803689, 5.48663785842827454468993325298, 6.98575817588523826030094619981, 8.084770912488636432540306005294, 8.935750786446224320152773933875, 9.781399486796132184227161639125

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