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Properties

Label 2-1134-63.59-c1-0-30
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 − 0.717·5-s + (−1 − 2.44i)7-s + 0.999i·8-s + (−0.621 − 0.358i)10-s − 3i·11-s + (−2.12 − 1.22i)13-s + (0.358 − 2.62i)14-s + (−0.5 + 0.866i)16-s + (2.95 − 5.12i)17-s + (−5.12 + 2.95i)19-s + (−0.358 − 0.621i)20-s + (1.5 − 2.59i)22-s − 4.24i·23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.320·5-s + (−0.377 − 0.925i)7-s + 0.353i·8-s + (−0.196 − 0.113i)10-s − 0.904i·11-s + (−0.588 − 0.339i)13-s + (0.0958 − 0.700i)14-s + (−0.125 + 0.216i)16-s + (0.717 − 1.24i)17-s + (−1.17 + 0.678i)19-s + (−0.0802 − 0.138i)20-s + (0.319 − 0.553i)22-s − 0.884i·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

\(L(1)\) \(\approx\) \(1.455209828\)
\(L(\frac12)\) \(\approx\) \(1.455209828\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2.44i)T \)
good5 \( 1 + 0.717T + 5T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 + (2.12 + 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.12 - 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.24iT - 23T^{2} \)
29 \( 1 + (6.27 - 3.62i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.86 + 4.54i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.121 - 0.210i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.91 + 10.2i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.121 - 0.210i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.95 + 5.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.27 - 3.62i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.03 + 6.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.878 + 0.507i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 + (1.24 + 0.717i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.37 + 2.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.31 - 5.74i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.7 - 6.77i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.730957898532118878756756486223, −8.623165760698465073699127161065, −7.76208164433711387068801871702, −7.18032441943151656110850417105, −6.21450690319229865194302362390, −5.41215605452539359275166973583, −4.28585909384322545360491359543, −3.60684981512190593862992057633, −2.50911725625774417764983743211, −0.49554967855078850422066594096, 1.79434272403426456215033364830, 2.72737636194934561065808009161, 3.91645280064645024538819632154, 4.69408629465646464645730636272, 5.76139356198985900959417051589, 6.43399708022167938853607043447, 7.48372860386284172603974102636, 8.344962052839841302017644997587, 9.444195616715772509893701615511, 9.914795814439582677368530842043

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