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Properties

Label 2-936-9.7-c1-0-25
Degree $2$
Conductor $936$
Sign $-0.266 + 0.963i$
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  + (−0.793 − 1.53i)3-s + (0.671 + 1.16i)5-s + (2.29 − 3.98i)7-s + (−1.74 + 2.44i)9-s + (0.873 − 1.51i)11-s + (−0.5 − 0.866i)13-s + (1.25 − 1.95i)15-s + 0.936·17-s + 1.28·19-s + (−7.95 − 0.378i)21-s + (0.711 + 1.23i)23-s + (1.59 − 2.76i)25-s + (5.14 + 0.739i)27-s + (−1.89 + 3.28i)29-s + (−3.71 − 6.43i)31-s + ⋯
L(s)  = 1  + (−0.458 − 0.888i)3-s + (0.300 + 0.520i)5-s + (0.869 − 1.50i)7-s + (−0.580 + 0.814i)9-s + (0.263 − 0.456i)11-s + (−0.138 − 0.240i)13-s + (0.324 − 0.505i)15-s + 0.227·17-s + 0.295·19-s + (−1.73 − 0.0826i)21-s + (0.148 + 0.256i)23-s + (0.319 − 0.553i)25-s + (0.989 + 0.142i)27-s + (−0.352 + 0.610i)29-s + (−0.667 − 1.15i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869330 - 1.14222i\)
\(L(\frac12)\) \(\approx\) \(0.869330 - 1.14222i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.793 + 1.53i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.671 - 1.16i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.29 + 3.98i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.873 + 1.51i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.936T + 17T^{2} \)
19 \( 1 - 1.28T + 19T^{2} \)
23 \( 1 + (-0.711 - 1.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.89 - 3.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.71 + 6.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.81T + 37T^{2} \)
41 \( 1 + (4.63 + 8.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.09 - 1.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.13 - 1.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.11T + 53T^{2} \)
59 \( 1 + (-0.361 - 0.626i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.607 + 1.05i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.54 + 4.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.34T + 71T^{2} \)
73 \( 1 + 15.9T + 73T^{2} \)
79 \( 1 + (-4.70 + 8.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.04 + 8.74i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + (4.56 - 7.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.13886214156580131051411916565, −8.837486954964441517504388064387, −7.75022050713866490950328413979, −7.38166133023084474158441925877, −6.50390644520274978060753912758, −5.58866053658057178717224899887, −4.56446253269558715513320656143, −3.35259962327964497239861808323, −1.91854730772275985281308253331, −0.75813660728091640594898891803, 1.61178769264283595394519327053, 2.94032196507401585094190844429, 4.33422600612281086517198280007, 5.17638242265693078434738438899, 5.60506027791632978653446951859, 6.70527781620013711256388974865, 8.051622403352503330430740677507, 8.928708904448290596754249316349, 9.314498265239257353094625307415, 10.21842068630486917504941353522

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