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Properties

Label 2-1183-7.2-c1-0-54
Degree $2$
Conductor $1183$
Sign $-0.638 + 0.769i$
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  + (0.787 − 1.36i)2-s + (−1.10 − 1.90i)3-s + (−0.239 − 0.414i)4-s + (−1.88 + 3.25i)5-s − 3.46·6-s + (0.609 − 2.57i)7-s + 2.39·8-s + (−0.924 + 1.60i)9-s + (2.96 + 5.12i)10-s + (1.19 + 2.07i)11-s + (−0.527 + 0.913i)12-s + (−3.03 − 2.85i)14-s + 8.28·15-s + (2.36 − 4.09i)16-s + (0.152 + 0.264i)17-s + (1.45 + 2.52i)18-s + ⋯
L(s)  = 1  + (0.556 − 0.964i)2-s + (−0.635 − 1.10i)3-s + (−0.119 − 0.207i)4-s + (−0.840 + 1.45i)5-s − 1.41·6-s + (0.230 − 0.973i)7-s + 0.846·8-s + (−0.308 + 0.533i)9-s + (0.936 + 1.62i)10-s + (0.361 + 0.625i)11-s + (−0.152 + 0.263i)12-s + (−0.809 − 0.763i)14-s + 2.13·15-s + (0.591 − 1.02i)16-s + (0.0369 + 0.0640i)17-s + (0.343 + 0.594i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.638 + 0.769i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.701506455\)
\(L(\frac12)\) \(\approx\) \(1.701506455\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.609 + 2.57i)T \)
13 \( 1 \)
good2 \( 1 + (-0.787 + 1.36i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.10 + 1.90i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.88 - 3.25i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.19 - 2.07i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.152 - 0.264i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.23 + 2.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.16 + 7.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.97T + 29T^{2} \)
31 \( 1 + (-1.82 - 3.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.82 + 4.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.27T + 41T^{2} \)
43 \( 1 + 9.84T + 43T^{2} \)
47 \( 1 + (-6.29 + 10.9i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.92 + 6.80i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.69 - 11.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.407 - 0.705i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.05 - 1.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.60T + 71T^{2} \)
73 \( 1 + (2.60 + 4.51i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.851 - 1.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.65T + 83T^{2} \)
89 \( 1 + (-6.05 + 10.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.06T + 97T^{2} \)
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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

−10.12221920880506219571797260840, −8.369536730372624347878665393125, −7.39393097911858728522103687009, −6.99248806237801918475214578224, −6.54561676017072465238998588037, −4.88919628773531947540961276214, −4.00754991503335719588218971058, −3.13858442845253139810232723265, −2.11718831100635975946476209417, −0.77478350136038007163908758443, 1.28818082695700627768865688673, 3.50616299729161408483878549266, 4.46825727147066776148659041316, 5.05147610952427058867949963127, 5.52468816510608168821002722354, 6.36647046252515233466814784023, 7.73438140734562121580822640390, 8.294688851110495622585360657834, 9.191956224383738510297646620349, 9.863857100900882500869792442342

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