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Properties

Label 2-1134-63.16-c1-0-10
Degree $2$
Conductor $1134$
Sign $-0.272 - 0.962i$
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 + 2.44i)7-s − 0.999·8-s + 4.24·11-s + (1.12 + 1.94i)13-s + (−1.62 + 2.09i)14-s + (−0.5 − 0.866i)16-s + (1.12 − 1.94i)19-s + (2.12 + 3.67i)22-s + 1.24·23-s − 5·25-s + (−1.12 + 1.94i)26-s + (−2.62 − 0.358i)28-s + (2.12 − 3.67i)29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.377 + 0.925i)7-s − 0.353·8-s + 1.27·11-s + (0.310 + 0.538i)13-s + (−0.433 + 0.558i)14-s + (−0.125 − 0.216i)16-s + (0.257 − 0.445i)19-s + (0.452 + 0.783i)22-s + 0.259·23-s − 25-s + (−0.219 + 0.380i)26-s + (−0.495 − 0.0677i)28-s + (0.393 − 0.682i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.272 - 0.962i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.272 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.033803902\)
\(L(\frac12)\) \(\approx\) \(2.033803902\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 2.44i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + (-1.12 - 1.94i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.12 + 1.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.24T + 23T^{2} \)
29 \( 1 + (-2.12 + 3.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.62 - 8.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.74 - 9.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.24 - 9.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.37 - 4.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.12 + 3.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.12 - 1.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.121 - 0.210i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.24T + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.378 + 0.655i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.12 + 14.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.74 + 9.94i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.24 - 3.88i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.704038184873077430596540722838, −9.137850333454099666841649611625, −8.431750585775864112319648987018, −7.53624803431177605232277384990, −6.52406943404557237396144454896, −5.98922418081981259733355221532, −4.93179569527938255578046857822, −4.11800526393798678339728959162, −2.97887351528297223984012302770, −1.58069310904531437747395411289, 0.875607131160451955133827264559, 2.01722857757644649047580231820, 3.61401955531830710289242884803, 3.99226403441310973087061179695, 5.17306533412580048168666889105, 6.09638356597467066913215482113, 7.07050301650544773169553773833, 7.923409333253073023356702622372, 8.933195111248893071095787755303, 9.706084780921555509583976216765

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