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Properties

Label 2-1134-63.4-c1-0-17
Degree $2$
Conductor $1134$
Sign $-0.0788 + 0.996i$
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 − 2·5-s + (−2.5 + 0.866i)7-s + 0.999·8-s + (1 − 1.73i)10-s + 5·11-s + (−3 + 5.19i)13-s + (0.500 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (2 − 3.46i)17-s + (2 + 3.46i)19-s + (0.999 + 1.73i)20-s + (−2.5 + 4.33i)22-s − 4·23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.894·5-s + (−0.944 + 0.327i)7-s + 0.353·8-s + (0.316 − 0.547i)10-s + 1.50·11-s + (−0.832 + 1.44i)13-s + (0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.485 − 0.840i)17-s + (0.458 + 0.794i)19-s + (0.223 + 0.387i)20-s + (−0.533 + 0.923i)22-s − 0.834·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.0788 + 0.996i$
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: \((2,\ 1134,\ (\ :1/2),\ -0.0788 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2520128454\)
\(L(\frac12)\) \(\approx\) \(0.2520128454\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + 2T + 5T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (6.5 - 11.2i)T + (-36.5 - 63.2i)T^{2} \)
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 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.448092749990224519202797238106, −8.950725681058421085899915573221, −7.77367562378114432082573297937, −7.15881380551124676361636732726, −6.42252269960447687974564183751, −5.54801338533978629810346687250, −4.18837361274079859330211458564, −3.67191903264812648699157957049, −1.97983036412156696180173622948, −0.13551756824583556318855705796, 1.27479362715653477529883469767, 3.06696002870834234948143484948, 3.58136113035547435633877093747, 4.56304648725065427304245471951, 5.86238034004201170196952722107, 6.92839724795108304249058462913, 7.62170484874550815547873658847, 8.466673374217054732087171258159, 9.359007501855332417732312408873, 10.04606473037796190443431056070

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