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Properties

Label 2-1134-63.16-c1-0-18
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 3.64·5-s + (1.32 − 2.29i)7-s + 0.999·8-s + (1.82 + 3.15i)10-s + 3.64·11-s + (2.32 + 4.02i)13-s − 2.64·14-s + (−0.5 − 0.866i)16-s + (−1.82 − 3.15i)17-s + (−1 + 1.73i)19-s + (1.82 − 3.15i)20-s + (−1.82 − 3.15i)22-s − 1.29·23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.63·5-s + (0.499 − 0.866i)7-s + 0.353·8-s + (0.576 + 0.998i)10-s + 1.09·11-s + (0.644 + 1.11i)13-s − 0.707·14-s + (−0.125 − 0.216i)16-s + (−0.442 − 0.765i)17-s + (−0.229 + 0.397i)19-s + (0.407 − 0.705i)20-s + (−0.388 − 0.673i)22-s − 0.269·23-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: $\chi_{1134} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.592 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8122343296\)
\(L(\frac12)\) \(\approx\) \(0.8122343296\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 2.29i)T \)
good5 \( 1 + 3.64T + 5T^{2} \)
11 \( 1 - 3.64T + 11T^{2} \)
13 \( 1 + (-2.32 - 4.02i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.82 + 3.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.29T + 23T^{2} \)
29 \( 1 + (1.17 - 2.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.32 + 4.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.96 + 10.3i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.46 + 9.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.46 + 4.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.17 - 7.23i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.67 + 6.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.14 + 1.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.6T + 71T^{2} \)
73 \( 1 + (5.29 + 9.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.61 + 9.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.64 + 11.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.46 - 4.27i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.79 - 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.406126800130242757832417271445, −8.767433458512497809976832409976, −7.916664225699552091919352545218, −7.26511669768213663603483403855, −6.47983844923273279297229680132, −4.71196693060035119321858425717, −4.02147117649571851346885479012, −3.56556178151201795095738223515, −1.81130086984003812497124771478, −0.47588489385711452381858145446, 1.23454390865304561705499369658, 3.07204009395523578727632319225, 4.11960352290582387787265735116, 4.91257124470191228577028689573, 6.11246885911679217265128999988, 6.77617035587372217733177584742, 8.059624011918499831477862375792, 8.174805290060708458277887158016, 8.925197322789979922414404168932, 9.998009713610486892001483170559

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