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Properties

Label 2-936-936.571-c0-0-5
Degree $2$
Conductor $936$
Sign $0.766 + 0.642i$
Analytic cond. $0.467124$
Root an. cond. $0.683465$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.173 − 0.984i)3-s + (−0.499 + 0.866i)4-s + (0.173 − 0.300i)5-s + (0.939 − 0.342i)6-s + (−0.939 − 1.62i)7-s − 0.999·8-s + (−0.939 − 0.342i)9-s + 0.347·10-s + (0.766 + 0.642i)12-s + (0.5 − 0.866i)13-s + (0.939 − 1.62i)14-s + (−0.266 − 0.223i)15-s + (−0.5 − 0.866i)16-s + 1.53·17-s + (−0.173 − 0.984i)18-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.173 − 0.984i)3-s + (−0.499 + 0.866i)4-s + (0.173 − 0.300i)5-s + (0.939 − 0.342i)6-s + (−0.939 − 1.62i)7-s − 0.999·8-s + (−0.939 − 0.342i)9-s + 0.347·10-s + (0.766 + 0.642i)12-s + (0.5 − 0.866i)13-s + (0.939 − 1.62i)14-s + (−0.266 − 0.223i)15-s + (−0.5 − 0.866i)16-s + 1.53·17-s + (−0.173 − 0.984i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.467124\)
Root analytic conductor: \(0.683465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145499386\)
\(L(\frac12)\) \(\approx\) \(1.145499386\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.173 + 0.984i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.53T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + 0.347T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.87T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−10.06506900870705766951354793276, −9.110145679098978791874479698929, −8.135368004430734219788789463738, −7.43885934550884356311291129598, −6.89903214080491936996079729513, −5.99532101080813594931681917486, −5.18244894728401172227236820279, −3.67665001129766546784627538157, −3.18856189737395708491296367380, −0.992581125609915865638302776907, 2.19880977026496456782097009518, 3.07314402366013536093382312096, 3.81801845966138062638154493539, 5.07795930920553080306164077079, 5.78702037269898557515899124224, 6.46894571734855944864049408598, 8.314646689570212762592143230686, 9.069243256547853761477605954659, 9.669485139608179540860869964529, 10.21133396582938225634760294742

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