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Properties

Label 2-1134-63.59-c1-0-25
Degree $2$
Conductor $1134$
Sign $0.164 + 0.986i$
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 − 2.44·5-s + (−1.62 + 2.09i)7-s + 0.999i·8-s + (−2.12 − 1.22i)10-s − 4.24i·11-s + (−3.62 − 2.09i)13-s + (−2.44 + 0.999i)14-s + (−0.5 + 0.866i)16-s + (1.22 − 2.12i)17-s + (4.24 − 2.44i)19-s + (−1.22 − 2.12i)20-s + (2.12 − 3.67i)22-s − 6i·23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 1.09·5-s + (−0.612 + 0.790i)7-s + 0.353i·8-s + (−0.670 − 0.387i)10-s − 1.27i·11-s + (−1.00 − 0.579i)13-s + (−0.654 + 0.267i)14-s + (−0.125 + 0.216i)16-s + (0.297 − 0.514i)17-s + (0.973 − 0.561i)19-s + (−0.273 − 0.474i)20-s + (0.452 − 0.783i)22-s − 1.25i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.164 + 0.986i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9535958903\)
\(L(\frac12)\) \(\approx\) \(0.9535958903\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.44T + 5T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 + (3.62 + 2.09i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.22 + 2.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.24 + 2.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (-8.87 + 5.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.86 - 2.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.62 - 2.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.22 - 2.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.97 - 6.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.15 + 1.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.22 + 2.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.621 - 0.358i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.74 - 3.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (13.2 + 7.64i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.62 - 8.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.74 + 4.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.87 + 15.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.74 - 3.31i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.550898113939878498507448569855, −8.550514502257400798142234930586, −7.987372235814744052939319788359, −7.08715753342019593403047306723, −6.22585238410655511724889225357, −5.32127534268325221281783539557, −4.50895219126279657021673636543, −3.20586065903968593582381807646, −2.83073500613375217699295336433, −0.34058452653445494379636304857, 1.52324145425655715529899315153, 3.03380757797395609866656792771, 3.91306851935200457229722900909, 4.54857889388651124801912255301, 5.56550874882377472344711684997, 7.00770309502175994788931696575, 7.20374091016987743206230135711, 8.151477797751983989570999638765, 9.661224566390216008127984738481, 9.858781415895023722257257213168

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