Properties

Label 2-1183-1.1-c1-0-41
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.381·2-s − 0.381·3-s − 1.85·4-s − 0.381·5-s + 0.145·6-s − 7-s + 1.47·8-s − 2.85·9-s + 0.145·10-s + 4.85·11-s + 0.708·12-s + 0.381·14-s + 0.145·15-s + 3.14·16-s + 7.47·17-s + 1.09·18-s − 4.85·19-s + 0.708·20-s + 0.381·21-s − 1.85·22-s + 4.47·23-s − 0.562·24-s − 4.85·25-s + 2.23·27-s + 1.85·28-s − 4.09·29-s − 0.0557·30-s + ⋯
L(s)  = 1  − 0.270·2-s − 0.220·3-s − 0.927·4-s − 0.170·5-s + 0.0595·6-s − 0.377·7-s + 0.520·8-s − 0.951·9-s + 0.0461·10-s + 1.46·11-s + 0.204·12-s + 0.102·14-s + 0.0376·15-s + 0.786·16-s + 1.81·17-s + 0.256·18-s − 1.11·19-s + 0.158·20-s + 0.0833·21-s − 0.395·22-s + 0.932·23-s − 0.114·24-s − 0.970·25-s + 0.430·27-s + 0.350·28-s − 0.759·29-s − 0.0101·30-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 \( 1 + 0.381T + 2T^{2} \)
3 \( 1 + 0.381T + 3T^{2} \)
5 \( 1 + 0.381T + 5T^{2} \)
11 \( 1 - 4.85T + 11T^{2} \)
17 \( 1 - 7.47T + 17T^{2} \)
19 \( 1 + 4.85T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 4.09T + 29T^{2} \)
31 \( 1 + 8.70T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 5.23T + 41T^{2} \)
43 \( 1 + 7.56T + 43T^{2} \)
47 \( 1 + 2.23T + 47T^{2} \)
53 \( 1 - 8.23T + 53T^{2} \)
59 \( 1 + 2.23T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 0.708T + 67T^{2} \)
71 \( 1 + 8.18T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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

−9.242418006721246493072843423992, −8.725758535407408592816681592195, −7.87717089995583770443176939577, −6.88671388003764420764706554300, −5.85852497429691635704084424954, −5.19769500022047440090298103222, −3.92723195464011465965801762874, −3.35764240683341862665267275000, −1.48942761865667432654331216995, 0, 1.48942761865667432654331216995, 3.35764240683341862665267275000, 3.92723195464011465965801762874, 5.19769500022047440090298103222, 5.85852497429691635704084424954, 6.88671388003764420764706554300, 7.87717089995583770443176939577, 8.725758535407408592816681592195, 9.242418006721246493072843423992

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