Properties

Label 2-2151-1.1-c1-0-96
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 $1$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.22·2-s + 2.93·4-s + 0.0570·5-s − 3.64·7-s + 2.06·8-s + 0.126·10-s − 3.71·11-s − 0.736·13-s − 8.09·14-s − 1.27·16-s − 1.72·17-s − 0.753·19-s + 0.167·20-s − 8.25·22-s − 1.29·23-s − 4.99·25-s − 1.63·26-s − 10.6·28-s − 2.49·29-s + 6.31·31-s − 6.95·32-s − 3.83·34-s − 0.207·35-s − 1.11·37-s − 1.67·38-s + 0.118·40-s + 0.457·41-s + ⋯
L(s)  = 1  + 1.57·2-s + 1.46·4-s + 0.0255·5-s − 1.37·7-s + 0.730·8-s + 0.0400·10-s − 1.12·11-s − 0.204·13-s − 2.16·14-s − 0.317·16-s − 0.418·17-s − 0.172·19-s + 0.0374·20-s − 1.75·22-s − 0.270·23-s − 0.999·25-s − 0.320·26-s − 2.01·28-s − 0.462·29-s + 1.13·31-s − 1.22·32-s − 0.657·34-s − 0.0351·35-s − 0.182·37-s − 0.271·38-s + 0.0186·40-s + 0.0715·41-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: \(1\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
239 \( 1 - T \)
good2 \( 1 - 2.22T + 2T^{2} \)
5 \( 1 - 0.0570T + 5T^{2} \)
7 \( 1 + 3.64T + 7T^{2} \)
11 \( 1 + 3.71T + 11T^{2} \)
13 \( 1 + 0.736T + 13T^{2} \)
17 \( 1 + 1.72T + 17T^{2} \)
19 \( 1 + 0.753T + 19T^{2} \)
23 \( 1 + 1.29T + 23T^{2} \)
29 \( 1 + 2.49T + 29T^{2} \)
31 \( 1 - 6.31T + 31T^{2} \)
37 \( 1 + 1.11T + 37T^{2} \)
41 \( 1 - 0.457T + 41T^{2} \)
43 \( 1 + 3.49T + 43T^{2} \)
47 \( 1 - 6.66T + 47T^{2} \)
53 \( 1 - 0.214T + 53T^{2} \)
59 \( 1 - 7.80T + 59T^{2} \)
61 \( 1 + 3.44T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 - 9.97T + 71T^{2} \)
73 \( 1 + 2.09T + 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 - 9.92T + 83T^{2} \)
89 \( 1 + 0.440T + 89T^{2} \)
97 \( 1 + 7.68T + 97T^{2} \)
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

−8.672363998801677974289082252860, −7.61287433706771007976653974472, −6.81810783419695244804323275496, −6.09959380843388502794067929871, −5.52603280008222371023049195400, −4.59862996213378650155705500281, −3.78119295870544768717971258865, −2.95968876355688432363099676661, −2.25711911582769317092470742776, 0, 2.25711911582769317092470742776, 2.95968876355688432363099676661, 3.78119295870544768717971258865, 4.59862996213378650155705500281, 5.52603280008222371023049195400, 6.09959380843388502794067929871, 6.81810783419695244804323275496, 7.61287433706771007976653974472, 8.672363998801677974289082252860

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