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Properties

Label 2-1134-63.41-c1-0-8
Degree $2$
Conductor $1134$
Sign $0.513 - 0.858i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 + 1.5i)5-s + (0.5 − 2.59i)7-s + 0.999i·8-s − 1.73i·10-s + (−2.59 + 1.5i)11-s + (−3 − 1.73i)13-s + (0.866 + 2.5i)14-s + (−0.5 − 0.866i)16-s + 6.92·17-s + 1.73i·19-s + (0.866 + 1.49i)20-s + (1.5 − 2.59i)22-s + (2.59 + 1.5i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.387 + 0.670i)5-s + (0.188 − 0.981i)7-s + 0.353i·8-s − 0.547i·10-s + (−0.783 + 0.452i)11-s + (−0.832 − 0.480i)13-s + (0.231 + 0.668i)14-s + (−0.125 − 0.216i)16-s + 1.68·17-s + 0.397i·19-s + (0.193 + 0.335i)20-s + (0.319 − 0.553i)22-s + (0.541 + 0.312i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.513 - 0.858i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.513 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.005099883\)
\(L(\frac12)\) \(\approx\) \(1.005099883\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (0.866 - 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + (-2.59 - 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.19 - 3i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.5 - 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + (-6.06 + 10.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.73 - 3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + (1.73 - 3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3iT - 71T^{2} \)
73 \( 1 + 3.46iT - 73T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.66 - 15i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5.19T + 89T^{2} \)
97 \( 1 + (-12 + 6.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.05838773297342475265886109679, −9.248210811847241738802885423230, −7.86177983655840960091934782919, −7.60207005838875188550028730750, −7.03418399690251759340898120916, −5.76545083040654029480937705635, −4.94478165139916874673703700370, −3.67990058248785797011976299119, −2.65749417175467379978049526952, −1.02842906320136551675501132810, 0.68766237499812458945480432282, 2.24320054594658711130895688567, 3.13512592337517155366068589247, 4.53310312398938953251370954156, 5.33621199539565006588941389433, 6.30501807027488483119562506812, 7.68718844698489152214917263022, 8.013973249210988721490422812015, 8.928594671585408426524726950958, 9.598598832231776603375750180186

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