Properties

Label 2-1183-91.55-c0-0-0
Degree $2$
Conductor $1183$
Sign $0.997 - 0.0743i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.623 − 1.07i)2-s + (−0.277 + 0.480i)4-s + (−0.5 + 0.866i)7-s − 0.554·8-s + (−0.5 + 0.866i)9-s + (0.222 + 0.385i)11-s + 1.24·14-s + (0.623 + 1.07i)16-s + 1.24·18-s + (0.277 − 0.480i)22-s + (0.900 + 1.56i)23-s + 25-s + (−0.277 − 0.480i)28-s + (−0.623 − 1.07i)29-s + (0.500 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.623 − 1.07i)2-s + (−0.277 + 0.480i)4-s + (−0.5 + 0.866i)7-s − 0.554·8-s + (−0.5 + 0.866i)9-s + (0.222 + 0.385i)11-s + 1.24·14-s + (0.623 + 1.07i)16-s + 1.24·18-s + (0.277 − 0.480i)22-s + (0.900 + 1.56i)23-s + 25-s + (−0.277 − 0.480i)28-s + (−0.623 − 1.07i)29-s + (0.500 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.997 - 0.0743i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ 0.997 - 0.0743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6046428884\)
\(L(\frac12)\) \(\approx\) \(0.6046428884\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.80T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.745303281706286347218317692668, −9.515001597879942265988664774985, −8.579084329668545966896331796351, −7.84625411006289113529584138965, −6.59909551827337460004370783247, −5.73892012087324760170508769521, −4.81967325135694429343913774321, −3.30633538852180643121480777003, −2.62741868770318602662041303307, −1.55758082602634536846886157221, 0.67460351321824723772964645058, 2.91549534127026354251162891927, 3.75403825897156642798294605088, 5.08854729075310595449900778446, 6.18126054697442499452007278869, 6.71558713175034376079722134442, 7.33464813278295762641199874663, 8.375117864087706295052881642736, 8.988437338629833400006900637370, 9.582822544027801345901615440034

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