Properties

Label 2-2151-1.1-c1-0-65
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  − 2.41·2-s + 3.82·4-s + 0.666·5-s + 0.549·7-s − 4.39·8-s − 1.60·10-s − 0.262·11-s + 3.05·13-s − 1.32·14-s + 2.96·16-s − 2.00·17-s + 1.22·19-s + 2.54·20-s + 0.633·22-s − 6.88·23-s − 4.55·25-s − 7.36·26-s + 2.10·28-s − 5.64·29-s + 2.28·31-s + 1.63·32-s + 4.84·34-s + 0.366·35-s + 8.46·37-s − 2.95·38-s − 2.93·40-s − 1.90·41-s + ⋯
L(s)  = 1  − 1.70·2-s + 1.91·4-s + 0.298·5-s + 0.207·7-s − 1.55·8-s − 0.508·10-s − 0.0791·11-s + 0.846·13-s − 0.354·14-s + 0.741·16-s − 0.487·17-s + 0.280·19-s + 0.569·20-s + 0.135·22-s − 1.43·23-s − 0.911·25-s − 1.44·26-s + 0.397·28-s − 1.04·29-s + 0.409·31-s + 0.289·32-s + 0.831·34-s + 0.0619·35-s + 1.39·37-s − 0.478·38-s − 0.463·40-s − 0.298·41-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 \( 1 + 2.41T + 2T^{2} \)
5 \( 1 - 0.666T + 5T^{2} \)
7 \( 1 - 0.549T + 7T^{2} \)
11 \( 1 + 0.262T + 11T^{2} \)
13 \( 1 - 3.05T + 13T^{2} \)
17 \( 1 + 2.00T + 17T^{2} \)
19 \( 1 - 1.22T + 19T^{2} \)
23 \( 1 + 6.88T + 23T^{2} \)
29 \( 1 + 5.64T + 29T^{2} \)
31 \( 1 - 2.28T + 31T^{2} \)
37 \( 1 - 8.46T + 37T^{2} \)
41 \( 1 + 1.90T + 41T^{2} \)
43 \( 1 + 7.39T + 43T^{2} \)
47 \( 1 + 1.67T + 47T^{2} \)
53 \( 1 + 7.14T + 53T^{2} \)
59 \( 1 + 3.67T + 59T^{2} \)
61 \( 1 + 13.7T + 61T^{2} \)
67 \( 1 + 2.80T + 67T^{2} \)
71 \( 1 - 4.78T + 71T^{2} \)
73 \( 1 - 2.16T + 73T^{2} \)
79 \( 1 + 5.81T + 79T^{2} \)
83 \( 1 - 3.57T + 83T^{2} \)
89 \( 1 - 9.71T + 89T^{2} \)
97 \( 1 - 7.62T + 97T^{2} \)
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

−8.741671670381373548560104459817, −7.956372622931938618507974984826, −7.59587780397416252352409446651, −6.38587652996124349552323134082, −6.02882479294191178168119734389, −4.66736969148240188359787186016, −3.47044337798719837657948493295, −2.18687580979108669513210263165, −1.45320558711694358504103237491, 0, 1.45320558711694358504103237491, 2.18687580979108669513210263165, 3.47044337798719837657948493295, 4.66736969148240188359787186016, 6.02882479294191178168119734389, 6.38587652996124349552323134082, 7.59587780397416252352409446651, 7.956372622931938618507974984826, 8.741671670381373548560104459817

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