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Properties

Label 2-1134-9.7-c1-0-22
Degree $2$
Conductor $1134$
Sign $-0.984 - 0.173i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.86 − 3.23i)5-s + (0.5 − 0.866i)7-s − 0.999·8-s − 3.73·10-s + (2.09 − 3.63i)11-s + (−0.232 − 0.401i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 7·17-s − 2.73·19-s + (−1.86 + 3.23i)20-s + (−2.09 − 3.63i)22-s + (−3.09 − 5.36i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.834 − 1.44i)5-s + (0.188 − 0.327i)7-s − 0.353·8-s − 1.18·10-s + (0.632 − 1.09i)11-s + (−0.0643 − 0.111i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s + 1.69·17-s − 0.626·19-s + (−0.417 + 0.722i)20-s + (−0.447 − 0.774i)22-s + (−0.645 − 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.330360747\)
\(L(\frac12)\) \(\approx\) \(1.330360747\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (1.86 + 3.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.09 + 3.63i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.232 + 0.401i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 + 2.73T + 19T^{2} \)
23 \( 1 + (3.09 + 5.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.23 - 7.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.09 - 1.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.66T + 37T^{2} \)
41 \( 1 + (4.73 + 8.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.73 - 4.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.633 - 1.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.53T + 53T^{2} \)
59 \( 1 + (-3.09 - 5.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.96 + 8.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.63 - 2.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + (7.56 - 13.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.29 + 12.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.92T + 89T^{2} \)
97 \( 1 + (-1.46 + 2.53i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.334483855178477064370761896834, −8.480001008386101999149281320921, −8.137688732539367183296138913353, −6.86051906035809528771133714065, −5.61497502576386169949745503667, −4.98651056292367953502996650695, −3.95207843706092304167718450726, −3.38393939780238345551199140049, −1.54555749821710919977399146638, −0.54149728045920001634620874394, 2.10469111124698932635189815323, 3.43272112536154252208592572490, 3.97980651813358106223305815823, 5.20618077169820451972536281678, 6.22928616918537220063514105044, 6.95130547672491767199069962486, 7.67176657694933762586987079887, 8.186302594504030730838717151879, 9.598906382388098566385347514588, 10.07927074220974619698786381073

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