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Properties

Label 2-1134-63.47-c1-0-21
Degree $2$
Conductor $1134$
Sign $-0.592 + 0.805i$
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 − 3.46·5-s + (0.5 + 2.59i)7-s − 0.999i·8-s + (−2.99 + 1.73i)10-s + (4.5 − 2.59i)13-s + (1.73 + 2i)14-s + (−0.5 − 0.866i)16-s + (−3.46 − 6i)17-s + (−3 − 1.73i)19-s + (−1.73 + 2.99i)20-s − 6i·23-s + 6.99·25-s + (2.59 − 4.5i)26-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 1.54·5-s + (0.188 + 0.981i)7-s − 0.353i·8-s + (−0.948 + 0.547i)10-s + (1.24 − 0.720i)13-s + (0.462 + 0.534i)14-s + (−0.125 − 0.216i)16-s + (−0.840 − 1.45i)17-s + (−0.688 − 0.397i)19-s + (−0.387 + 0.670i)20-s − 1.25i·23-s + 1.39·25-s + (0.509 − 0.882i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.190036896\)
\(L(\frac12)\) \(\approx\) \(1.190036896\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + 3.46T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.46 + 6i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.5 + 4.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.46 - 6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.46 + 6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.46 + 6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.5 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (6 - 3.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.5 - 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.73 - 3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.427095903277877988712977205523, −8.623319586535743228143334034369, −7.989918490097758025291763077333, −6.96378026123602187245090680843, −6.07144734353037755321345358304, −5.00546586301644104064392790759, −4.24466287493884740053332112215, −3.30555717565926101217848575539, −2.34541692072223609313900296505, −0.43116114068380460526076196448, 1.57988232738071881661164918240, 3.61852164188572521199129828852, 3.85162777668728726353291850999, 4.64152600345882242733972865123, 6.00457091086607445018291338383, 6.83115991368779958834534872470, 7.57128651422368079937134207489, 8.279197761597880811433880799532, 8.931791716947998120246091090413, 10.43832550214922127347035578719

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