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Properties

Label 2-1134-63.20-c1-0-31
Degree $2$
Conductor $1134$
Sign $-0.884 + 0.466i$
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 + (−1.22 − 2.12i)5-s + (−1.62 − 2.09i)7-s + 0.999i·8-s − 2.44i·10-s + (−2.12 + 1.22i)13-s + (−0.358 − 2.62i)14-s + (−0.5 + 0.866i)16-s − 4.89·17-s + 2.44i·19-s + (1.22 − 2.12i)20-s + (−5.19 + 3i)23-s + (−0.499 + 0.866i)25-s − 2.44·26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.547 − 0.948i)5-s + (−0.612 − 0.790i)7-s + 0.353i·8-s − 0.774i·10-s + (−0.588 + 0.339i)13-s + (−0.0958 − 0.700i)14-s + (−0.125 + 0.216i)16-s − 1.18·17-s + 0.561i·19-s + (0.273 − 0.474i)20-s + (−1.08 + 0.625i)23-s + (−0.0999 + 0.173i)25-s − 0.480·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.884 + 0.466i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.884 + 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4446583821\)
\(L(\frac12)\) \(\approx\) \(0.4446583821\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 + 2.09i)T \)
good5 \( 1 + (1.22 + 2.12i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.12 - 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.19 + 3i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (2.44 + 4.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.44 + 4.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + (6.12 + 10.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.6 - 6.12i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.22 + 2.12i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (4.24 + 2.44i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.422536545459517989245725531272, −8.508039384730777038219241022927, −7.71828790781931875492284338429, −6.96295312426284071270129011122, −6.08711995059775196159161794350, −5.02205861897428358283871755660, −4.20803664220089564087435087609, −3.59103740701272334135111319919, −2.00410399304068148285485030774, −0.14411384948269391431524818713, 2.21959246135466579939384074822, 2.95289864226857084958278396510, 3.89481167934856389867900410432, 4.94834566293787616694668314485, 5.98606353441294201682904610794, 6.73095997601504953127209855535, 7.45480095883011927692482118370, 8.621939411737943897104188234317, 9.457439111645383257161249151642, 10.35977824162747809611944411436

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