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Properties

Label 2-1134-63.59-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 + 1.73·5-s + (2 + 1.73i)7-s − 0.999i·8-s + (−1.49 − 0.866i)10-s − 3i·11-s + (−3 − 1.73i)13-s + (−0.866 − 2.5i)14-s + (−0.5 + 0.866i)16-s + (1.73 − 3i)17-s + (3 − 1.73i)19-s + (0.866 + 1.49i)20-s + (−1.5 + 2.59i)22-s − 6i·23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 0.774·5-s + (0.755 + 0.654i)7-s − 0.353i·8-s + (−0.474 − 0.273i)10-s − 0.904i·11-s + (−0.832 − 0.480i)13-s + (−0.231 − 0.668i)14-s + (−0.125 + 0.216i)16-s + (0.420 − 0.727i)17-s + (0.688 − 0.397i)19-s + (0.193 + 0.335i)20-s + (−0.319 + 0.553i)22-s − 1.25i·23-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: $\chi_{1134} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.462437224\)
\(L(\frac12)\) \(\approx\) \(1.462437224\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.73 + 3i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + (-2.59 + 1.5i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.46 + 6i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.46 - 6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.79 - 4.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.33 + 7.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.5 + 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.623619819789008499888661409022, −9.018135487922843582518970236609, −8.145230847403672945674992825757, −7.48228336043529762884240887681, −6.26687113198860589411384510313, −5.49290891046844509228026847495, −4.58407989606545191798028235804, −2.96615293646324386216206869263, −2.31848280304992018754178618450, −0.873439987737102373229381500782, 1.35454178472985072128998066348, 2.20846608875597257820842495388, 3.87723588150726914368281782721, 4.99547342732678598709836581906, 5.69890620798806598302165388949, 6.84903794746048614098365437719, 7.47202424044890210173647578294, 8.180935929824875208213158762575, 9.277429536584714555808168608050, 9.942023006038742215346829888613

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