Properties

Label 2-2151-1.1-c1-0-12
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.88·2-s + 1.55·4-s − 1.57·5-s − 2.47·7-s + 0.845·8-s + 2.97·10-s + 3.83·11-s − 3.18·13-s + 4.65·14-s − 4.69·16-s + 4.51·17-s + 4.54·19-s − 2.44·20-s − 7.22·22-s − 1.92·23-s − 2.51·25-s + 6.00·26-s − 3.83·28-s + 0.0587·29-s − 9.58·31-s + 7.15·32-s − 8.49·34-s + 3.89·35-s + 0.789·37-s − 8.56·38-s − 1.33·40-s + 2.07·41-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.775·4-s − 0.705·5-s − 0.934·7-s + 0.298·8-s + 0.939·10-s + 1.15·11-s − 0.883·13-s + 1.24·14-s − 1.17·16-s + 1.09·17-s + 1.04·19-s − 0.547·20-s − 1.54·22-s − 0.401·23-s − 0.502·25-s + 1.17·26-s − 0.724·28-s + 0.0109·29-s − 1.72·31-s + 1.26·32-s − 1.45·34-s + 0.658·35-s + 0.129·37-s − 1.38·38-s − 0.210·40-s + 0.324·41-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

\(L(1)\) \(\approx\) \(0.5219440589\)
\(L(\frac12)\) \(\approx\) \(0.5219440589\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.88T + 2T^{2} \)
5 \( 1 + 1.57T + 5T^{2} \)
7 \( 1 + 2.47T + 7T^{2} \)
11 \( 1 - 3.83T + 11T^{2} \)
13 \( 1 + 3.18T + 13T^{2} \)
17 \( 1 - 4.51T + 17T^{2} \)
19 \( 1 - 4.54T + 19T^{2} \)
23 \( 1 + 1.92T + 23T^{2} \)
29 \( 1 - 0.0587T + 29T^{2} \)
31 \( 1 + 9.58T + 31T^{2} \)
37 \( 1 - 0.789T + 37T^{2} \)
41 \( 1 - 2.07T + 41T^{2} \)
43 \( 1 - 2.55T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 0.844T + 53T^{2} \)
59 \( 1 - 1.09T + 59T^{2} \)
61 \( 1 - 2.97T + 61T^{2} \)
67 \( 1 - 1.93T + 67T^{2} \)
71 \( 1 - 8.02T + 71T^{2} \)
73 \( 1 + 6.42T + 73T^{2} \)
79 \( 1 - 8.18T + 79T^{2} \)
83 \( 1 - 9.82T + 83T^{2} \)
89 \( 1 + 8.85T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.440668773715265489690457920842, −8.314760434658418845365595737978, −7.58555514265721617886941286280, −7.14175415448134590041552031271, −6.23154950600642281171492532411, −5.13355871632369959164809991789, −3.95972934251439923979662392231, −3.25115955239197695019260462690, −1.80957575256963323366859808270, −0.58186875671990591098887996610, 0.58186875671990591098887996610, 1.80957575256963323366859808270, 3.25115955239197695019260462690, 3.95972934251439923979662392231, 5.13355871632369959164809991789, 6.23154950600642281171492532411, 7.14175415448134590041552031271, 7.58555514265721617886941286280, 8.314760434658418845365595737978, 9.440668773715265489690457920842

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