Properties

Label 2-1134-63.41-c1-0-16
Degree $2$
Conductor $1134$
Sign $0.0341 + 0.999i$
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 + (−1.23 + 2.13i)5-s + (−1.03 + 2.43i)7-s + 0.999i·8-s − 2.46i·10-s + (−2.27 + 1.31i)11-s + (−2.35 − 1.35i)13-s + (−0.320 − 2.62i)14-s + (−0.5 − 0.866i)16-s − 4.00·17-s − 8.27i·19-s + (1.23 + 2.13i)20-s + (1.31 − 2.27i)22-s + (1.39 + 0.807i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.550 + 0.953i)5-s + (−0.391 + 0.920i)7-s + 0.353i·8-s − 0.778i·10-s + (−0.686 + 0.396i)11-s + (−0.652 − 0.376i)13-s + (−0.0856 − 0.701i)14-s + (−0.125 − 0.216i)16-s − 0.972·17-s − 1.89i·19-s + (0.275 + 0.476i)20-s + (0.280 − 0.485i)22-s + (0.291 + 0.168i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0341 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0341 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.0341 + 0.999i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.0341 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1624783875\)
\(L(\frac12)\) \(\approx\) \(0.1624783875\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.03 - 2.43i)T \)
good5 \( 1 + (1.23 - 2.13i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.27 - 1.31i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.35 + 1.35i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.00T + 17T^{2} \)
19 \( 1 + 8.27iT - 19T^{2} \)
23 \( 1 + (-1.39 - 0.807i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.71 - 0.992i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.737 - 0.425i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 11.9T + 37T^{2} \)
41 \( 1 + (-2.76 + 4.78i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.61 + 6.25i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.90 - 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.19iT - 53T^{2} \)
59 \( 1 + (-0.773 + 1.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.57 - 4.37i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.230 + 0.399i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.30iT - 71T^{2} \)
73 \( 1 + 0.0359iT - 73T^{2} \)
79 \( 1 + (0.700 + 1.21i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.06 + 13.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + (-1.56 + 0.901i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.419008111981854603481231481013, −8.935697194438760906454090021440, −7.83157945091050343876151985245, −7.17643270161107768343384878412, −6.53411749401828295059341293775, −5.47887457193378915418128456801, −4.53522445243372996475300474879, −2.92253485687050346538547500378, −2.40947099092654019930607663200, −0.094195536625100030746116560419, 1.17645882069965356281134362815, 2.65693510659582357931605233456, 3.95674818709583499285380584515, 4.54228322470897445113603648951, 5.85023815370095869461659658536, 6.91001578343995826843483551919, 7.88082830237737194521554447878, 8.236252007336315906702700515532, 9.306414115975422190632230903253, 9.938073016761964709618681031916

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