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Properties

Label 2-1183-7.4-c1-0-57
Degree $2$
Conductor $1183$
Sign $0.999 + 0.00429i$
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  + (0.308 + 0.534i)2-s + (−1.53 + 2.66i)3-s + (0.809 − 1.40i)4-s + (−0.698 − 1.20i)5-s − 1.89·6-s + (2.36 + 1.18i)7-s + 2.23·8-s + (−3.22 − 5.58i)9-s + (0.431 − 0.746i)10-s + (0.111 − 0.193i)11-s + (2.48 + 4.30i)12-s + (0.0964 + 1.63i)14-s + 4.29·15-s + (−0.928 − 1.60i)16-s + (1.10 − 1.90i)17-s + (1.98 − 3.44i)18-s + ⋯
L(s)  = 1  + (0.218 + 0.378i)2-s + (−0.887 + 1.53i)3-s + (0.404 − 0.700i)4-s + (−0.312 − 0.540i)5-s − 0.774·6-s + (0.894 + 0.447i)7-s + 0.790·8-s + (−1.07 − 1.86i)9-s + (0.136 − 0.236i)10-s + (0.0337 − 0.0584i)11-s + (0.718 + 1.24i)12-s + (0.0257 + 0.435i)14-s + 1.10·15-s + (−0.232 − 0.402i)16-s + (0.267 − 0.462i)17-s + (0.469 − 0.812i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.999 + 0.00429i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 0.999 + 0.00429i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.417309096\)
\(L(\frac12)\) \(\approx\) \(1.417309096\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.36 - 1.18i)T \)
13 \( 1 \)
good2 \( 1 + (-0.308 - 0.534i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.53 - 2.66i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.698 + 1.20i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.111 + 0.193i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.10 + 1.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.07 + 7.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.72 + 4.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.94T + 29T^{2} \)
31 \( 1 + (2.36 - 4.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.859 + 1.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.09T + 41T^{2} \)
43 \( 1 + 2.39T + 43T^{2} \)
47 \( 1 + (2.34 + 4.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.61 + 9.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.743 - 1.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.25 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.845 + 1.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.64T + 71T^{2} \)
73 \( 1 + (2.25 - 3.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.78 - 8.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.77T + 83T^{2} \)
89 \( 1 + (-5.36 - 9.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−10.07221576167684521450314466353, −8.894424359662879444427767313051, −8.463451021287783294253068032646, −6.96525120552828428730620779850, −6.18623067710695463405781762017, −5.20949781529912627868430863030, −4.81237926500209907870252950379, −4.21938547102378023098802088706, −2.53552667771469235059909820519, −0.67937561826021860560993600386, 1.38025645872881428289999652471, 2.11901108762544315568264584237, 3.46803326727054459148452878386, 4.53829011169206686719066871362, 5.79845407894725291210772594649, 6.52960918975930071286678972776, 7.40969275041777457815892642236, 7.81549420946385398490344191546, 8.420221339600766977740612302553, 10.31642767889634379756326712995

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