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Properties

Label 2-1134-9.2-c2-0-14
Degree $2$
Conductor $1134$
Sign $-0.766 - 0.642i$
Analytic cond. $30.8992$
Root an. cond. $5.55871$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−5.25 + 3.03i)5-s + (1.32 − 2.29i)7-s + 2.82i·8-s − 8.58·10-s + (10.5 + 6.06i)11-s + (9.29 + 16.0i)13-s + (3.24 − 1.87i)14-s + (−2.00 + 3.46i)16-s − 10.9i·17-s + 20·19-s + (−10.5 − 6.06i)20-s + (8.58 + 14.8i)22-s + (−10.5 + 6.06i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.05 + 0.606i)5-s + (0.188 − 0.327i)7-s + 0.353i·8-s − 0.858·10-s + (0.955 + 0.551i)11-s + (0.714 + 1.23i)13-s + (0.231 − 0.133i)14-s + (−0.125 + 0.216i)16-s − 0.641i·17-s + 1.05·19-s + (−0.525 − 0.303i)20-s + (0.390 + 0.675i)22-s + (−0.457 + 0.263i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.766 - 0.642i$
Analytic conductor: \(30.8992\)
Root analytic conductor: \(5.55871\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1),\ -0.766 - 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.973181269\)
\(L(\frac12)\) \(\approx\) \(1.973181269\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (-1.32 + 2.29i)T \)
good5 \( 1 + (5.25 - 3.03i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-10.5 - 6.06i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-9.29 - 16.0i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 10.9iT - 289T^{2} \)
19 \( 1 - 20T + 361T^{2} \)
23 \( 1 + (10.5 - 6.06i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (36.2 + 20.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (12.5 + 21.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 38T + 1.36e3T^{2} \)
41 \( 1 + (52.5 - 30.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (41.7 - 72.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-14.6 - 8.48i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 94.0iT - 2.80e3T^{2} \)
59 \( 1 + (50.4 - 29.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (7.83 - 13.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-66.3 - 114. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 12.1iT - 5.04e3T^{2} \)
73 \( 1 + 76.9T + 5.32e3T^{2} \)
79 \( 1 + (16.8 - 29.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-52.4 - 30.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 4.77iT - 7.92e3T^{2} \)
97 \( 1 + (-94.2 + 163. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.815229020844801925259029778646, −9.146361595734851520366345176424, −7.945053086677766188258838848018, −7.36757213266105069909006431770, −6.71989009741101679066160811746, −5.81635645849818878788674001791, −4.42826221517221682894694470670, −4.02563555060268598043930092539, −3.05741416085387142968091723842, −1.51818351425576642480727469984, 0.50318410622064081490544760211, 1.69157879732901682335398933332, 3.52661284838254713035230324895, 3.63128406596415461472847613455, 5.00500376546058893608130061920, 5.65460817757587973564298936771, 6.67916300730420018359229204853, 7.79973292977030824461376503568, 8.457470739583731503509527794383, 9.193079335885673158052074266923

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