Properties

Label 2-1183-1.1-c1-0-31
Degree $2$
Conductor $1183$
Sign $1$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.35·2-s + 3.39·3-s − 0.155·4-s − 0.772·5-s − 4.61·6-s + 7-s + 2.92·8-s + 8.54·9-s + 1.04·10-s − 2.11·11-s − 0.527·12-s − 1.35·14-s − 2.62·15-s − 3.66·16-s + 5.63·17-s − 11.6·18-s + 2.99·19-s + 0.119·20-s + 3.39·21-s + 2.87·22-s + 1.24·23-s + 9.94·24-s − 4.40·25-s + 18.8·27-s − 0.155·28-s − 7.96·29-s + 3.56·30-s + ⋯
L(s)  = 1  − 0.960·2-s + 1.96·3-s − 0.0776·4-s − 0.345·5-s − 1.88·6-s + 0.377·7-s + 1.03·8-s + 2.84·9-s + 0.331·10-s − 0.638·11-s − 0.152·12-s − 0.362·14-s − 0.677·15-s − 0.916·16-s + 1.36·17-s − 2.73·18-s + 0.688·19-s + 0.0268·20-s + 0.741·21-s + 0.612·22-s + 0.258·23-s + 2.03·24-s − 0.880·25-s + 3.62·27-s − 0.0293·28-s − 1.47·29-s + 0.650·30-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.901118807\)
\(L(\frac12)\) \(\approx\) \(1.901118807\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
13 \( 1 \)
good2 \( 1 + 1.35T + 2T^{2} \)
3 \( 1 - 3.39T + 3T^{2} \)
5 \( 1 + 0.772T + 5T^{2} \)
11 \( 1 + 2.11T + 11T^{2} \)
17 \( 1 - 5.63T + 17T^{2} \)
19 \( 1 - 2.99T + 19T^{2} \)
23 \( 1 - 1.24T + 23T^{2} \)
29 \( 1 + 7.96T + 29T^{2} \)
31 \( 1 - 1.26T + 31T^{2} \)
37 \( 1 + 4.54T + 37T^{2} \)
41 \( 1 - 9.82T + 41T^{2} \)
43 \( 1 - 2.64T + 43T^{2} \)
47 \( 1 - 7.68T + 47T^{2} \)
53 \( 1 + 0.350T + 53T^{2} \)
59 \( 1 + 5.00T + 59T^{2} \)
61 \( 1 - 4.24T + 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 - 8.99T + 73T^{2} \)
79 \( 1 + 6.30T + 79T^{2} \)
83 \( 1 - 2.92T + 83T^{2} \)
89 \( 1 + 1.97T + 89T^{2} \)
97 \( 1 - 7.88T + 97T^{2} \)
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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

−9.472383286943046092783774282206, −9.011639069426055083315493694338, −8.029787829080322750536234341403, −7.75231746328340505450154821100, −7.23826948243562887842797351121, −5.37832318857272289885173091574, −4.22019023286483094146177473575, −3.46755824899646798917931106334, −2.32371169146141955987718912529, −1.22282062218325438912849796505, 1.22282062218325438912849796505, 2.32371169146141955987718912529, 3.46755824899646798917931106334, 4.22019023286483094146177473575, 5.37832318857272289885173091574, 7.23826948243562887842797351121, 7.75231746328340505450154821100, 8.029787829080322750536234341403, 9.011639069426055083315493694338, 9.472383286943046092783774282206

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