Properties

Label 2-1183-1.1-c1-0-71
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.10·2-s + 2.26·3-s + 2.44·4-s + 3.60·5-s + 4.77·6-s − 7-s + 0.948·8-s + 2.11·9-s + 7.59·10-s − 0.886·11-s + 5.53·12-s − 2.10·14-s + 8.14·15-s − 2.89·16-s − 4.96·17-s + 4.46·18-s − 2.37·19-s + 8.82·20-s − 2.26·21-s − 1.86·22-s − 3.85·23-s + 2.14·24-s + 7.97·25-s − 2.00·27-s − 2.44·28-s + 1.28·29-s + 17.1·30-s + ⋯
L(s)  = 1  + 1.49·2-s + 1.30·3-s + 1.22·4-s + 1.61·5-s + 1.94·6-s − 0.377·7-s + 0.335·8-s + 0.705·9-s + 2.40·10-s − 0.267·11-s + 1.59·12-s − 0.563·14-s + 2.10·15-s − 0.724·16-s − 1.20·17-s + 1.05·18-s − 0.545·19-s + 1.97·20-s − 0.493·21-s − 0.398·22-s − 0.804·23-s + 0.437·24-s + 1.59·25-s − 0.385·27-s − 0.462·28-s + 0.238·29-s + 3.13·30-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: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.122641452\)
\(L(\frac12)\) \(\approx\) \(6.122641452\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.10T + 2T^{2} \)
3 \( 1 - 2.26T + 3T^{2} \)
5 \( 1 - 3.60T + 5T^{2} \)
11 \( 1 + 0.886T + 11T^{2} \)
17 \( 1 + 4.96T + 17T^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
23 \( 1 + 3.85T + 23T^{2} \)
29 \( 1 - 1.28T + 29T^{2} \)
31 \( 1 + 8.46T + 31T^{2} \)
37 \( 1 - 9.63T + 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 + 3.64T + 43T^{2} \)
47 \( 1 + 2.98T + 47T^{2} \)
53 \( 1 - 4.92T + 53T^{2} \)
59 \( 1 - 7.32T + 59T^{2} \)
61 \( 1 + 1.53T + 61T^{2} \)
67 \( 1 - 8.42T + 67T^{2} \)
71 \( 1 - 6.44T + 71T^{2} \)
73 \( 1 - 7.14T + 73T^{2} \)
79 \( 1 - 0.757T + 79T^{2} \)
83 \( 1 - 4.76T + 83T^{2} \)
89 \( 1 + 3.61T + 89T^{2} \)
97 \( 1 - 0.463T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.485036434398529755912030396543, −9.198478430162745129361601201322, −8.181768305404457817453594329326, −6.96251237394552441394811344261, −6.17017924137132380167718346023, −5.55570376213944457495759096774, −4.45534354027615169845676364917, −3.57408339232078730579409503325, −2.40082592635421042784848442429, −2.21168817826102840035626114450, 2.21168817826102840035626114450, 2.40082592635421042784848442429, 3.57408339232078730579409503325, 4.45534354027615169845676364917, 5.55570376213944457495759096774, 6.17017924137132380167718346023, 6.96251237394552441394811344261, 8.181768305404457817453594329326, 9.198478430162745129361601201322, 9.485036434398529755912030396543

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