Properties

Label 2-1183-1.1-c1-0-29
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.93·2-s − 2.54·3-s + 1.74·4-s − 0.312·5-s + 4.92·6-s + 7-s + 0.495·8-s + 3.48·9-s + 0.603·10-s − 4.16·11-s − 4.43·12-s − 1.93·14-s + 0.794·15-s − 4.44·16-s − 5.20·17-s − 6.73·18-s + 4.87·19-s − 0.544·20-s − 2.54·21-s + 8.06·22-s + 3.39·23-s − 1.26·24-s − 4.90·25-s − 1.22·27-s + 1.74·28-s + 3.54·29-s − 1.53·30-s + ⋯
L(s)  = 1  − 1.36·2-s − 1.46·3-s + 0.871·4-s − 0.139·5-s + 2.01·6-s + 0.377·7-s + 0.175·8-s + 1.16·9-s + 0.190·10-s − 1.25·11-s − 1.28·12-s − 0.517·14-s + 0.205·15-s − 1.11·16-s − 1.26·17-s − 1.58·18-s + 1.11·19-s − 0.121·20-s − 0.555·21-s + 1.71·22-s + 0.708·23-s − 0.257·24-s − 0.980·25-s − 0.235·27-s + 0.329·28-s + 0.658·29-s − 0.280·30-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: \(1\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.93T + 2T^{2} \)
3 \( 1 + 2.54T + 3T^{2} \)
5 \( 1 + 0.312T + 5T^{2} \)
11 \( 1 + 4.16T + 11T^{2} \)
17 \( 1 + 5.20T + 17T^{2} \)
19 \( 1 - 4.87T + 19T^{2} \)
23 \( 1 - 3.39T + 23T^{2} \)
29 \( 1 - 3.54T + 29T^{2} \)
31 \( 1 - 9.52T + 31T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 - 0.433T + 41T^{2} \)
43 \( 1 + 8.96T + 43T^{2} \)
47 \( 1 + 8.62T + 47T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 4.87T + 61T^{2} \)
67 \( 1 + 8.71T + 67T^{2} \)
71 \( 1 + 6.09T + 71T^{2} \)
73 \( 1 - 3.59T + 73T^{2} \)
79 \( 1 - 9.90T + 79T^{2} \)
83 \( 1 - 0.500T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 2.35T + 97T^{2} \)
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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

−9.603435432087333285556767525256, −8.425602558274234850093291128031, −7.86212312229427124607852549566, −6.97126681586404320921988536780, −6.18332889222660899740147899254, −5.07311972145462707238561508820, −4.53978455244479268350370192070, −2.60239125648269313273355711268, −1.13138530065947532124486780832, 0, 1.13138530065947532124486780832, 2.60239125648269313273355711268, 4.53978455244479268350370192070, 5.07311972145462707238561508820, 6.18332889222660899740147899254, 6.97126681586404320921988536780, 7.86212312229427124607852549566, 8.425602558274234850093291128031, 9.603435432087333285556767525256

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