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Properties

Label 2-1183-7.2-c1-0-91
Degree $2$
Conductor $1183$
Sign $0.820 - 0.571i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
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  + (1.37 − 2.37i)2-s + (−1.20 − 2.09i)3-s + (−2.76 − 4.78i)4-s + (0.312 − 0.540i)5-s − 6.61·6-s + (−1.27 + 2.31i)7-s − 9.65·8-s + (−1.41 + 2.44i)9-s + (−0.856 − 1.48i)10-s + (−1.23 − 2.13i)11-s + (−6.66 + 11.5i)12-s + (3.76 + 6.20i)14-s − 1.50·15-s + (−7.72 + 13.3i)16-s + (0.00903 + 0.0156i)17-s + (3.87 + 6.70i)18-s + ⋯
L(s)  = 1  + (0.969 − 1.67i)2-s + (−0.696 − 1.20i)3-s + (−1.38 − 2.39i)4-s + (0.139 − 0.241i)5-s − 2.70·6-s + (−0.480 + 0.876i)7-s − 3.41·8-s + (−0.470 + 0.815i)9-s + (−0.270 − 0.468i)10-s + (−0.371 − 0.644i)11-s + (−1.92 + 3.33i)12-s + (1.00 + 1.65i)14-s − 0.389·15-s + (−1.93 + 3.34i)16-s + (0.00219 + 0.00379i)17-s + (0.913 + 1.58i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.820 - 0.571i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.820 - 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.005989887\)
\(L(\frac12)\) \(\approx\) \(1.005989887\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.27 - 2.31i)T \)
13 \( 1 \)
good2 \( 1 + (-1.37 + 2.37i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.312 + 0.540i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.23 + 2.13i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.00903 - 0.0156i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.67 + 4.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.66 + 2.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.10T + 29T^{2} \)
31 \( 1 + (2.58 + 4.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.94 - 3.37i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.60T + 41T^{2} \)
43 \( 1 - 0.158T + 43T^{2} \)
47 \( 1 + (-0.732 + 1.26i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.41 + 2.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.25 - 7.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.86 - 8.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0205 + 0.0356i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.95T + 71T^{2} \)
73 \( 1 + (6.26 + 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.40 + 7.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + (-1.30 + 2.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.04T + 97T^{2} \)
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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

−9.226337393825216736299703797392, −8.562531590405413350175884849102, −7.00172863616775617348201744937, −6.11190537029928190529523703378, −5.48095169360143924791516048711, −4.78217917903717863777761891610, −3.29106140043383746416086377488, −2.55528247008785643695504355202, −1.45787567626025402870201170010, −0.35164515859898827537881187628, 3.24427641989202378998020911347, 3.92171012941868115656983016782, 4.78611066754149063273107805663, 5.32678917125462762003542472124, 6.23654861080769060335404292993, 6.97991415909946891361547277809, 7.68796384241240204574684625074, 8.688550476404057572090075670952, 9.776303707005388384007773260810, 10.21150955796736444991225281216

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