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Properties

Label 2-936-117.94-c1-0-12
Degree $2$
Conductor $936$
Sign $0.892 - 0.450i$
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  + (−0.983 − 1.42i)3-s + (−0.702 + 1.21i)5-s − 2.51·7-s + (−1.06 + 2.80i)9-s + (1.01 − 1.76i)11-s + (0.489 − 3.57i)13-s + (2.42 − 0.194i)15-s + (−3.08 + 5.34i)17-s + (0.189 − 0.328i)19-s + (2.47 + 3.58i)21-s + 8.73·23-s + (1.51 + 2.62i)25-s + (5.04 − 1.23i)27-s + (−1.04 + 1.81i)29-s + (−2.09 + 3.62i)31-s + ⋯
L(s)  = 1  + (−0.567 − 0.823i)3-s + (−0.314 + 0.543i)5-s − 0.949·7-s + (−0.355 + 0.934i)9-s + (0.306 − 0.530i)11-s + (0.135 − 0.990i)13-s + (0.626 − 0.0502i)15-s + (−0.748 + 1.29i)17-s + (0.0435 − 0.0754i)19-s + (0.539 + 0.781i)21-s + 1.82·23-s + (0.302 + 0.524i)25-s + (0.971 − 0.237i)27-s + (−0.194 + 0.337i)29-s + (−0.376 + 0.651i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.892 - 0.450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.892 - 0.450i$
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: \((2,\ 936,\ (\ :1/2),\ 0.892 - 0.450i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.887680 + 0.211053i\)
\(L(\frac12)\) \(\approx\) \(0.887680 + 0.211053i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.983 + 1.42i)T \)
13 \( 1 + (-0.489 + 3.57i)T \)
good5 \( 1 + (0.702 - 1.21i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.51T + 7T^{2} \)
11 \( 1 + (-1.01 + 1.76i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.08 - 5.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.189 + 0.328i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.73T + 23T^{2} \)
29 \( 1 + (1.04 - 1.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.09 - 3.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.87 - 8.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.57T + 41T^{2} \)
43 \( 1 - 1.60T + 43T^{2} \)
47 \( 1 + (0.0950 + 0.164i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + (-3.80 - 6.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 7.59T + 61T^{2} \)
67 \( 1 - 3.81T + 67T^{2} \)
71 \( 1 + (-5.34 + 9.25i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.57T + 73T^{2} \)
79 \( 1 + (5.34 + 9.26i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.605 - 1.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.08 - 7.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−10.51073217681533124011377796528, −9.177367175571984240907463984518, −8.371330910235564588917328956765, −7.39413109362712746767936736761, −6.65586950942643180066368305551, −6.06434819868132882909094159865, −5.05864168715777361796713665212, −3.56403419244089974001593820082, −2.74147508067386546411987594536, −1.05066884334917137470419013023, 0.57929918585128782365365550367, 2.65158675793137705732867676847, 3.95383629189523429760629186285, 4.57059635036551553948185867778, 5.52639068392244015701627127278, 6.62663704179198246228202313785, 7.17676974635360594793114933314, 8.707069464735516490814930742866, 9.387086546742273308712137714014, 9.683713213968085884896388786719

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