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Properties

Label 2-1184-296.147-c0-0-2
Degree $2$
Conductor $1184$
Sign $1$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61·3-s − 0.618·5-s + 1.61·9-s − 0.618·11-s + 1.61·13-s − 1.00·15-s − 1.61·23-s − 0.618·25-s + 27-s + 1.61·29-s + 0.618·31-s − 1.00·33-s − 37-s + 2.61·39-s − 1.61·41-s − 1.00·45-s + 49-s + 0.381·55-s − 0.618·61-s − 1.00·65-s − 0.618·67-s − 2.61·69-s + 0.618·73-s − 0.999·75-s − 1.61·79-s − 2·83-s + 2.61·87-s + ⋯
L(s)  = 1  + 1.61·3-s − 0.618·5-s + 1.61·9-s − 0.618·11-s + 1.61·13-s − 1.00·15-s − 1.61·23-s − 0.618·25-s + 27-s + 1.61·29-s + 0.618·31-s − 1.00·33-s − 37-s + 2.61·39-s − 1.61·41-s − 1.00·45-s + 49-s + 0.381·55-s − 0.618·61-s − 1.00·65-s − 0.618·67-s − 2.61·69-s + 0.618·73-s − 0.999·75-s − 1.61·79-s − 2·83-s + 2.61·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1184\)    =    \(2^{5} \cdot 37\)
Sign: $1$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1184} (591, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1184,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.618333454\)
\(L(\frac12)\) \(\approx\) \(1.618333454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + T \)
good3 \( 1 - 1.61T + T^{2} \)
5 \( 1 + 0.618T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 0.618T + T^{2} \)
13 \( 1 - 1.61T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 - 1.61T + T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
41 \( 1 + 1.61T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 + 0.618T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
79 \( 1 + 1.61T + T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.968219034448885951442668954150, −8.790561210111846589270321201821, −8.354266605023914429546947348860, −7.88064426810386678526792029352, −6.87636165876594307903175770707, −5.83156639375107175493975019924, −4.39598853167543765742219262920, −3.68336620594756791501151567073, −2.89120194444196721567905865826, −1.69653568636025266031598342389, 1.69653568636025266031598342389, 2.89120194444196721567905865826, 3.68336620594756791501151567073, 4.39598853167543765742219262920, 5.83156639375107175493975019924, 6.87636165876594307903175770707, 7.88064426810386678526792029352, 8.354266605023914429546947348860, 8.790561210111846589270321201821, 9.968219034448885951442668954150

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