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Properties

Label 2-1134-9.7-c1-0-23
Degree $2$
Conductor $1134$
Sign $-0.766 - 0.642i$
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 − 1.73i)5-s + (0.5 − 0.866i)7-s − 0.999·8-s − 1.99·10-s + (−2 + 3.46i)11-s + (−3 − 5.19i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 2·17-s − 4·19-s + (−0.999 + 1.73i)20-s + (1.99 + 3.46i)22-s + (4 + 6.92i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.447 − 0.774i)5-s + (0.188 − 0.327i)7-s − 0.353·8-s − 0.632·10-s + (−0.603 + 1.04i)11-s + (−0.832 − 1.44i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s − 0.485·17-s − 0.917·19-s + (−0.223 + 0.387i)20-s + (0.426 + 0.738i)22-s + (0.834 + 1.44i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.766 - 0.642i$
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: \((2,\ 1134,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5256622570\)
\(L(\frac12)\) \(\approx\) \(0.5256622570\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2 - 3.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.414940994046001405079471365323, −8.560338148998537164405704964220, −7.68800240946826680593727913074, −6.96370913749017170345196380992, −5.42631066592115025936896753413, −4.97060345755627881930025798668, −4.08321840651976725638466863125, −2.94731104322690226353970298812, −1.73703919101765279046127405963, −0.19229586220280067418553044688, 2.28836752178602409335801169070, 3.25438324421925786745887771407, 4.39605801767808140083202818606, 5.14018480024727090237070710375, 6.42619518965534409040733330419, 6.76432306362789556021490776602, 7.76977622800345579708962766913, 8.589052375645920029969326396141, 9.225156556763939278066647661870, 10.49490286720802973267046654617

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