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Properties

Label 2-1134-21.5-c1-0-29
Degree $2$
Conductor $1134$
Sign $-0.694 + 0.719i$
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 + (1.82 − 3.15i)5-s + (−1.04 − 2.43i)7-s − 0.999i·8-s + (−3.15 + 1.82i)10-s + (4.38 − 2.53i)11-s − 3.39i·13-s + (−0.310 + 2.62i)14-s + (−0.5 + 0.866i)16-s + (0.774 + 1.34i)17-s + (−0.707 − 0.408i)19-s + 3.64·20-s − 5.06·22-s + (1.47 + 0.850i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.814 − 1.41i)5-s + (−0.394 − 0.918i)7-s − 0.353i·8-s + (−0.997 + 0.576i)10-s + (1.32 − 0.763i)11-s − 0.942i·13-s + (−0.0829 + 0.702i)14-s + (−0.125 + 0.216i)16-s + (0.187 + 0.325i)17-s + (−0.162 − 0.0936i)19-s + 0.814·20-s − 1.08·22-s + (0.307 + 0.177i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.694 + 0.719i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.694 + 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.363646948\)
\(L(\frac12)\) \(\approx\) \(1.363646948\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.04 + 2.43i)T \)
good5 \( 1 + (-1.82 + 3.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.38 + 2.53i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.39iT - 13T^{2} \)
17 \( 1 + (-0.774 - 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.707 + 0.408i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.47 - 0.850i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.16iT - 29T^{2} \)
31 \( 1 + (-1.87 + 1.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.39 - 5.88i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.03T + 41T^{2} \)
43 \( 1 + 6.12T + 43T^{2} \)
47 \( 1 + (3.37 - 5.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.4 + 6.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.08 + 1.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.28 - 3.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.22 + 2.12i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.74iT - 71T^{2} \)
73 \( 1 + (-3.76 + 2.17i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.37 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.53T + 83T^{2} \)
89 \( 1 + (-6.01 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.46iT - 97T^{2} \)
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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

−9.603164021001687318602312505094, −8.679279410884712435826126873305, −8.293609386828978736802009281872, −7.03963797875601418683356346067, −6.18755698841218423897451651772, −5.24342137824587487373454210089, −4.13820023230270475458365156611, −3.18479237834535183246706941090, −1.50335378199694477909762918554, −0.78649522100574018455393474614, 1.81283363127403922384824665575, 2.58443019615818561970158499760, 3.86616230121560733795598258440, 5.30677595084518369598400007395, 6.37874359774594553710056730822, 6.61802865327963926493607181051, 7.41199054308508908577839349789, 8.750147326903297542135259971849, 9.370790483215377024624496689780, 9.904481863556233706609255598190

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