Properties

Label 2-1134-63.4-c1-0-30
Degree $2$
Conductor $1134$
Sign $-0.975 - 0.220i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 5-s + (2 − 1.73i)7-s − 0.999·8-s + (−0.5 + 0.866i)10-s − 5·11-s + (−0.499 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (−2 + 3.46i)17-s + (−4 − 6.92i)19-s + (0.499 + 0.866i)20-s + (−2.5 + 4.33i)22-s + 4·23-s − 4·25-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (0.755 − 0.654i)7-s − 0.353·8-s + (−0.158 + 0.273i)10-s − 1.50·11-s + (−0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.485 + 0.840i)17-s + (−0.917 − 1.58i)19-s + (0.111 + 0.193i)20-s + (−0.533 + 0.923i)22-s + 0.834·23-s − 0.800·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.975 - 0.220i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7373549650\)
\(L(\frac12)\) \(\approx\) \(0.7373549650\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + T + 5T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (2.5 + 4.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.5 + 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.5 - 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.5 + 6.06i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.5 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.539724846620569600839619587997, −8.455097941681831741228575242327, −7.85306818704118945017810025630, −6.94158105914127859257004100464, −5.78255094467497370465565471221, −4.73545741747289084490952996941, −4.25235097662830384124164097681, −2.97184717759455502003556207532, −1.93831245664780331247692642475, −0.26708513409587534597085687755, 2.04739439191801017158504118707, 3.20587130788720384140379930745, 4.40424962104469368739024281006, 5.22168614633782176726790053995, 5.84497064628903122383256838000, 7.05555626716470911464647289123, 7.83576576311369556179647744827, 8.350707281662787806307649899356, 9.193709751118144522342267709656, 10.33561829894124781275833330163

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