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Properties

Label 2-936-117.94-c1-0-2
Degree $2$
Conductor $936$
Sign $-0.509 + 0.860i$
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  + (0.0896 + 1.72i)3-s + (−1.74 + 3.02i)5-s − 3.26·7-s + (−2.98 + 0.310i)9-s + (−2.07 + 3.59i)11-s + (3.60 + 0.120i)13-s + (−5.39 − 2.75i)15-s + (0.552 − 0.957i)17-s + (2.87 − 4.98i)19-s + (−0.292 − 5.63i)21-s + 0.808·23-s + (−3.60 − 6.24i)25-s + (−0.804 − 5.13i)27-s + (−0.545 + 0.945i)29-s + (0.736 − 1.27i)31-s + ⋯
L(s)  = 1  + (0.0517 + 0.998i)3-s + (−0.781 + 1.35i)5-s − 1.23·7-s + (−0.994 + 0.103i)9-s + (−0.626 + 1.08i)11-s + (0.999 + 0.0334i)13-s + (−1.39 − 0.710i)15-s + (0.134 − 0.232i)17-s + (0.660 − 1.14i)19-s + (−0.0637 − 1.23i)21-s + 0.168·23-s + (−0.720 − 1.24i)25-s + (−0.154 − 0.987i)27-s + (−0.101 + 0.175i)29-s + (0.132 − 0.229i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.509 + 0.860i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (913, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.509 + 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211249 - 0.370482i\)
\(L(\frac12)\) \(\approx\) \(0.211249 - 0.370482i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0896 - 1.72i)T \)
13 \( 1 + (-3.60 - 0.120i)T \)
good5 \( 1 + (1.74 - 3.02i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.26T + 7T^{2} \)
11 \( 1 + (2.07 - 3.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.552 + 0.957i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.87 + 4.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.808T + 23T^{2} \)
29 \( 1 + (0.545 - 0.945i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.736 + 1.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.26 - 5.64i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.64T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 + (-0.872 - 1.51i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.33T + 53T^{2} \)
59 \( 1 + (1.70 + 2.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + 9.98T + 67T^{2} \)
71 \( 1 + (7.73 - 13.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.48T + 73T^{2} \)
79 \( 1 + (6.20 + 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.53 - 2.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.25 + 5.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.51T + 97T^{2} \)
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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

−10.34539164486540815582553660984, −10.05413180237120304118036542966, −9.145088343191139155409897450305, −8.114414500672056554954257089358, −7.06094039715688898899051486935, −6.54446751148453365786513791128, −5.32648797011141855904433999040, −4.20435115329743594438461403741, −3.26805739701886048851810007249, −2.76690517964210519032166823317, 0.21038343020330968426874895774, 1.32504481639329446414382741847, 3.11841050067883467277300437258, 3.83158643712500552719473597535, 5.40793622695838313833932695496, 5.95918733655196390732556033437, 6.98992054246463438395840151484, 8.107157289123249631147953467816, 8.400623886219725037217016643144, 9.230603035975793871069261414801

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