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Properties

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

\(L(\frac{1}{2})\) \(\approx\) \(1.050104762\)
\(L(\frac12)\) \(\approx\) \(1.050104762\)
\(L(1)\) 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 \( 1 + 1.61T + T^{2} \)
7 \( 1 + 0.618T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.61T + T^{2} \)
17 \( 1 - 0.618T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
31 \( 1 + 0.618T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.618T + T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
61 \( 1 - 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
79 \( 1 - 1.61T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + 1.61T + T^{2} \)
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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

−10.61538254760137338434849738194, −9.319628248660324443625541919409, −8.026497732542470937651210227160, −7.07178375486281940218648856737, −6.48277511298224935502289049105, −5.69465854576682597227114274798, −4.97309202629132727401051446891, −3.95095855178449911771354271405, −3.25267216476052250460876334024, −1.15959764657448848261523418261, 1.15959764657448848261523418261, 3.25267216476052250460876334024, 3.95095855178449911771354271405, 4.97309202629132727401051446891, 5.69465854576682597227114274798, 6.48277511298224935502289049105, 7.07178375486281940218648856737, 8.026497732542470937651210227160, 9.319628248660324443625541919409, 10.61538254760137338434849738194

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