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Properties

Label 2-1134-7.2-c1-0-17
Degree $2$
Conductor $1134$
Sign $0.757 + 0.653i$
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 + (0.880 − 1.52i)5-s + (2.56 + 0.658i)7-s − 0.999·8-s + (−0.880 − 1.52i)10-s + (3.06 + 5.30i)11-s + 0.760·13-s + (1.85 − 1.88i)14-s + (−0.5 + 0.866i)16-s + (3.42 + 5.92i)17-s + (0.971 − 1.68i)19-s − 1.76·20-s + 6.12·22-s + (−0.210 + 0.364i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.393 − 0.681i)5-s + (0.968 + 0.249i)7-s − 0.353·8-s + (−0.278 − 0.482i)10-s + (0.923 + 1.59i)11-s + 0.211·13-s + (0.494 − 0.505i)14-s + (−0.125 + 0.216i)16-s + (0.829 + 1.43i)17-s + (0.222 − 0.385i)19-s − 0.393·20-s + 1.30·22-s + (−0.0438 + 0.0760i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.757 + 0.653i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.757 + 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.436350050\)
\(L(\frac12)\) \(\approx\) \(2.436350050\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 - 0.658i)T \)
good5 \( 1 + (-0.880 + 1.52i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.06 - 5.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.760T + 13T^{2} \)
17 \( 1 + (-3.42 - 5.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.971 + 1.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.210 - 0.364i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.46T + 29T^{2} \)
31 \( 1 + (3.85 + 6.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.44 + 2.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.94T + 41T^{2} \)
43 \( 1 + 8.66T + 43T^{2} \)
47 \( 1 + (-0.830 + 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.112 - 0.195i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.993 - 1.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.17 + 8.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.39 + 5.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (-0.153 - 0.265i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.12T + 83T^{2} \)
89 \( 1 + (1.30 - 2.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.63T + 97T^{2} \)
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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

−9.752024688323455469160190880670, −9.069131766374499481897233377279, −8.248042249337915885439188774576, −7.27050268492457246879618855024, −6.13043938737937165302244007878, −5.21689185501827017839741352555, −4.53252501907953445090545105112, −3.63938356121400714879875100340, −1.96404690994354780882859979109, −1.44979093194400235010572495937, 1.19271664493966280017416656540, 2.91067094641259578958686924251, 3.70181631041236373328971388603, 4.95042465566032147423620772093, 5.68458322720539355061254974141, 6.58400647022600455545709098858, 7.26151252620628026419661055910, 8.283904402364239237912381906309, 8.814914029912856319780932789102, 9.921497283400957197132213759035

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