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Properties

Label 2-936-936.259-c0-0-3
Degree $2$
Conductor $936$
Sign $0.173 + 0.984i$
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 + (0.766 − 0.642i)3-s + (−0.499 − 0.866i)4-s + (0.766 + 1.32i)5-s + (−0.173 − 0.984i)6-s + (0.173 − 0.300i)7-s − 0.999·8-s + (0.173 − 0.984i)9-s + 1.53·10-s + (−0.939 − 0.342i)12-s + (0.5 + 0.866i)13-s + (−0.173 − 0.300i)14-s + (1.43 + 0.524i)15-s + (−0.5 + 0.866i)16-s − 1.87·17-s + (−0.766 − 0.642i)18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.766 − 0.642i)3-s + (−0.499 − 0.866i)4-s + (0.766 + 1.32i)5-s + (−0.173 − 0.984i)6-s + (0.173 − 0.300i)7-s − 0.999·8-s + (0.173 − 0.984i)9-s + 1.53·10-s + (−0.939 − 0.342i)12-s + (0.5 + 0.866i)13-s + (−0.173 − 0.300i)14-s + (1.43 + 0.524i)15-s + (−0.5 + 0.866i)16-s − 1.87·17-s + (−0.766 − 0.642i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(0.467124\)
Root analytic conductor: \(0.683465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :0),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.669009365\)
\(L(\frac12)\) \(\approx\) \(1.669009365\)
\(L(1)\) 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 \( 1 + (-0.766 + 0.642i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + 1.87T + T^{2} \)
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 \( 1 + 1.53T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 0.347T + T^{2} \)
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

−10.23103262159873894386315506852, −9.264576352207110235489033930428, −8.761876564877716050713368707085, −7.30639573699005427488699116700, −6.59353433252468939667676209800, −5.96570693415535447880588695325, −4.40104435265045100063904526627, −3.49317464137807264296066886381, −2.42352033014093584883473888023, −1.81834806125956389067996948600, 2.06493405684043981181533529892, 3.40633640256392816317199152755, 4.54001301664613502700015127109, 5.09266895522690221936234168107, 5.89701763050257807392212344778, 7.05567871107031819010618893597, 8.291496048935321125587524812559, 8.682585368851903913511732261380, 9.189247743143536352258125867563, 10.16094061877368684262317111471

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