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Properties

Label 2-1134-63.20-c1-0-6
Degree $2$
Conductor $1134$
Sign $0.934 + 0.356i$
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 + (−1.73 − 3i)5-s + (−2.5 + 0.866i)7-s − 0.999i·8-s + 3.46i·10-s + (5.19 + 3i)11-s + (−1.5 + 0.866i)13-s + (2.59 + 0.500i)14-s + (−0.5 + 0.866i)16-s − 1.73·17-s + 6.92i·19-s + (1.73 − 2.99i)20-s + (−3 − 5.19i)22-s + (2.59 − 1.5i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.774 − 1.34i)5-s + (−0.944 + 0.327i)7-s − 0.353i·8-s + 1.09i·10-s + (1.56 + 0.904i)11-s + (−0.416 + 0.240i)13-s + (0.694 + 0.133i)14-s + (−0.125 + 0.216i)16-s − 0.420·17-s + 1.58i·19-s + (0.387 − 0.670i)20-s + (−0.639 − 1.10i)22-s + (0.541 − 0.312i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.934 + 0.356i$
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: \((2,\ 1134,\ (\ :1/2),\ 0.934 + 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8924584612\)
\(L(\frac12)\) \(\approx\) \(0.8924584612\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (1.73 + 3i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.19 - 3i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
23 \( 1 + (-2.59 + 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.59 - 1.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.5 + 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (3.46 + 6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.46 + 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + (-4.33 - 7.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12 - 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.73 + 3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.19T + 89T^{2} \)
97 \( 1 + (-6 - 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.698703469379646681989807563397, −8.798966559430989545806910176770, −8.596078914877383916169284553886, −7.31710477764034824197527814137, −6.68057058448482890671993454285, −5.47326716760131007573977296515, −4.22272551254325117153989942672, −3.75841836771058362514566126829, −2.12709260732711027798047961142, −0.866466918777364631281491154705, 0.70937677937849112180812066906, 2.77489431743503757169074812847, 3.42770597324321728871097104756, 4.57222185740111262567021036026, 6.19787768422033970859441032396, 6.67447731204549851154565640790, 7.16743279905089430822417670925, 8.179613171419237353835369226396, 9.131281612542155932888102251475, 9.727751672669730007921640089173

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