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Properties

Label 2-1160-1160.579-c0-0-7
Degree $2$
Conductor $1160$
Sign $1$
Analytic cond. $0.578915$
Root an. cond. $0.760864$
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  + 2-s + 0.618·3-s + 4-s − 5-s + 0.618·6-s + 1.61·7-s + 8-s − 0.618·9-s − 10-s + 0.618·12-s − 0.618·13-s + 1.61·14-s − 0.618·15-s + 16-s − 1.61·17-s − 0.618·18-s − 20-s + 1.00·21-s − 0.618·23-s + 0.618·24-s + 25-s − 0.618·26-s − 27-s + 1.61·28-s − 29-s − 0.618·30-s + 1.61·31-s + ⋯
L(s)  = 1  + 2-s + 0.618·3-s + 4-s − 5-s + 0.618·6-s + 1.61·7-s + 8-s − 0.618·9-s − 10-s + 0.618·12-s − 0.618·13-s + 1.61·14-s − 0.618·15-s + 16-s − 1.61·17-s − 0.618·18-s − 20-s + 1.00·21-s − 0.618·23-s + 0.618·24-s + 25-s − 0.618·26-s − 27-s + 1.61·28-s − 29-s − 0.618·30-s + 1.61·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1160\)    =    \(2^{3} \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(0.578915\)
Root analytic conductor: \(0.760864\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1160} (579, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1160,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.126784761\)
\(L(\frac12)\) \(\approx\) \(2.126784761\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.25297148081094490810686177902, −8.787478978343084718482855872707, −8.221363363011877229832579870045, −7.58744122980974681960875477848, −6.73584101265995587748568802544, −5.48041984461398223489674192397, −4.61487544549217139940640306946, −4.05929876473963900584595126359, −2.82884429279692423108327934738, −1.92480588808352345743983604656, 1.92480588808352345743983604656, 2.82884429279692423108327934738, 4.05929876473963900584595126359, 4.61487544549217139940640306946, 5.48041984461398223489674192397, 6.73584101265995587748568802544, 7.58744122980974681960875477848, 8.221363363011877229832579870045, 8.787478978343084718482855872707, 10.25297148081094490810686177902

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