Properties

Label 2-1183-1.1-c1-0-19
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  − 2.61·2-s − 2.61·3-s + 4.85·4-s − 2.61·5-s + 6.85·6-s − 7-s − 7.47·8-s + 3.85·9-s + 6.85·10-s − 1.85·11-s − 12.7·12-s + 2.61·14-s + 6.85·15-s + 9.85·16-s − 1.47·17-s − 10.0·18-s + 1.85·19-s − 12.7·20-s + 2.61·21-s + 4.85·22-s − 4.47·23-s + 19.5·24-s + 1.85·25-s − 2.23·27-s − 4.85·28-s + 7.09·29-s − 17.9·30-s + ⋯
L(s)  = 1  − 1.85·2-s − 1.51·3-s + 2.42·4-s − 1.17·5-s + 2.79·6-s − 0.377·7-s − 2.64·8-s + 1.28·9-s + 2.16·10-s − 0.559·11-s − 3.66·12-s + 0.699·14-s + 1.76·15-s + 2.46·16-s − 0.357·17-s − 2.37·18-s + 0.425·19-s − 2.84·20-s + 0.571·21-s + 1.03·22-s − 0.932·23-s + 3.99·24-s + 0.370·25-s − 0.430·27-s − 0.917·28-s + 1.31·29-s − 3.27·30-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: $\chi_{1183} (1, \cdot )$
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 \( 1 + 2.61T + 2T^{2} \)
3 \( 1 + 2.61T + 3T^{2} \)
5 \( 1 + 2.61T + 5T^{2} \)
11 \( 1 + 1.85T + 11T^{2} \)
17 \( 1 + 1.47T + 17T^{2} \)
19 \( 1 - 1.85T + 19T^{2} \)
23 \( 1 + 4.47T + 23T^{2} \)
29 \( 1 - 7.09T + 29T^{2} \)
31 \( 1 - 4.70T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 0.763T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 - 3.76T + 53T^{2} \)
59 \( 1 - 2.23T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + 4.90T + 89T^{2} \)
97 \( 1 + 18.8T + 97T^{2} \)
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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

−9.513819580291334317990998484185, −8.404871596259288604045417825207, −7.84298405200365218042406102391, −6.97939045327443668455989999066, −6.39601888460097042814218355910, −5.40272913042532002594487859668, −4.11636170255038719593876476212, −2.62223351766023596639712071955, −0.925409949379851576213774966540, 0, 0.925409949379851576213774966540, 2.62223351766023596639712071955, 4.11636170255038719593876476212, 5.40272913042532002594487859668, 6.39601888460097042814218355910, 6.97939045327443668455989999066, 7.84298405200365218042406102391, 8.404871596259288604045417825207, 9.513819580291334317990998484185

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