Properties

Label 2-2151-1.1-c1-0-42
Degree $2$
Conductor $2151$
Sign $1$
Analytic cond. $17.1758$
Root an. cond. $4.14437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 0.946·2-s − 1.10·4-s + 4.23·5-s + 2.19·7-s − 2.93·8-s + 4.00·10-s − 2.38·11-s − 5.96·13-s + 2.07·14-s − 0.570·16-s + 4.21·17-s + 5.07·19-s − 4.67·20-s − 2.25·22-s + 0.949·23-s + 12.9·25-s − 5.64·26-s − 2.42·28-s + 0.816·29-s + 9.04·31-s + 5.33·32-s + 3.99·34-s + 9.29·35-s + 9.83·37-s + 4.80·38-s − 12.4·40-s + 4.51·41-s + ⋯
L(s)  = 1  + 0.669·2-s − 0.552·4-s + 1.89·5-s + 0.829·7-s − 1.03·8-s + 1.26·10-s − 0.717·11-s − 1.65·13-s + 0.555·14-s − 0.142·16-s + 1.02·17-s + 1.16·19-s − 1.04·20-s − 0.480·22-s + 0.197·23-s + 2.58·25-s − 1.10·26-s − 0.458·28-s + 0.151·29-s + 1.62·31-s + 0.943·32-s + 0.684·34-s + 1.57·35-s + 1.61·37-s + 0.779·38-s − 1.96·40-s + 0.704·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2151\)    =    \(3^{2} \cdot 239\)
Sign: $1$
Analytic conductor: \(17.1758\)
Root analytic conductor: \(4.14437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2151} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.084590006\)
\(L(\frac12)\) \(\approx\) \(3.084590006\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
239 \( 1 - T \)
good2 \( 1 - 0.946T + 2T^{2} \)
5 \( 1 - 4.23T + 5T^{2} \)
7 \( 1 - 2.19T + 7T^{2} \)
11 \( 1 + 2.38T + 11T^{2} \)
13 \( 1 + 5.96T + 13T^{2} \)
17 \( 1 - 4.21T + 17T^{2} \)
19 \( 1 - 5.07T + 19T^{2} \)
23 \( 1 - 0.949T + 23T^{2} \)
29 \( 1 - 0.816T + 29T^{2} \)
31 \( 1 - 9.04T + 31T^{2} \)
37 \( 1 - 9.83T + 37T^{2} \)
41 \( 1 - 4.51T + 41T^{2} \)
43 \( 1 + 4.33T + 43T^{2} \)
47 \( 1 + 6.16T + 47T^{2} \)
53 \( 1 + 8.87T + 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 + 2.93T + 61T^{2} \)
67 \( 1 - 9.58T + 67T^{2} \)
71 \( 1 + 3.41T + 71T^{2} \)
73 \( 1 + 7.39T + 73T^{2} \)
79 \( 1 + 8.43T + 79T^{2} \)
83 \( 1 + 1.97T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 - 3.33T + 97T^{2} \)
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

−9.374224722667276259357958076773, −8.288481914131073261277109424216, −7.57065548480157639332843101899, −6.42479830784893832777491292786, −5.63464551050471377807840334734, −5.03825677026832100307037406151, −4.70253515189088944614633888172, −3.02281134511018921301406258528, −2.44101788409633765202696333194, −1.12771464448261886233080464413, 1.12771464448261886233080464413, 2.44101788409633765202696333194, 3.02281134511018921301406258528, 4.70253515189088944614633888172, 5.03825677026832100307037406151, 5.63464551050471377807840334734, 6.42479830784893832777491292786, 7.57065548480157639332843101899, 8.288481914131073261277109424216, 9.374224722667276259357958076773

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