Properties

Label 2-2151-1.1-c1-0-25
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  + 1.26·2-s − 0.402·4-s − 1.81·5-s + 1.66·7-s − 3.03·8-s − 2.29·10-s + 2.19·11-s + 3.06·13-s + 2.10·14-s − 3.03·16-s − 6.98·17-s + 8.13·19-s + 0.728·20-s + 2.76·22-s + 1.10·23-s − 1.71·25-s + 3.87·26-s − 0.670·28-s − 4.03·29-s − 1.76·31-s + 2.23·32-s − 8.82·34-s − 3.02·35-s + 11.5·37-s + 10.2·38-s + 5.50·40-s + 9.69·41-s + ⋯
L(s)  = 1  + 0.893·2-s − 0.201·4-s − 0.810·5-s + 0.630·7-s − 1.07·8-s − 0.724·10-s + 0.660·11-s + 0.850·13-s + 0.563·14-s − 0.758·16-s − 1.69·17-s + 1.86·19-s + 0.162·20-s + 0.590·22-s + 0.229·23-s − 0.343·25-s + 0.760·26-s − 0.126·28-s − 0.750·29-s − 0.316·31-s + 0.395·32-s − 1.51·34-s − 0.510·35-s + 1.89·37-s + 1.66·38-s + 0.869·40-s + 1.51·41-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2151,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.214109435\)
\(L(\frac12)\) \(\approx\) \(2.214109435\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.26T + 2T^{2} \)
5 \( 1 + 1.81T + 5T^{2} \)
7 \( 1 - 1.66T + 7T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 - 3.06T + 13T^{2} \)
17 \( 1 + 6.98T + 17T^{2} \)
19 \( 1 - 8.13T + 19T^{2} \)
23 \( 1 - 1.10T + 23T^{2} \)
29 \( 1 + 4.03T + 29T^{2} \)
31 \( 1 + 1.76T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 - 9.69T + 41T^{2} \)
43 \( 1 - 0.395T + 43T^{2} \)
47 \( 1 - 2.34T + 47T^{2} \)
53 \( 1 - 8.09T + 53T^{2} \)
59 \( 1 - 14.1T + 59T^{2} \)
61 \( 1 + 0.631T + 61T^{2} \)
67 \( 1 + 0.325T + 67T^{2} \)
71 \( 1 - 8.73T + 71T^{2} \)
73 \( 1 + 1.04T + 73T^{2} \)
79 \( 1 - 17.2T + 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + 6.30T + 89T^{2} \)
97 \( 1 - 3.04T + 97T^{2} \)
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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

−9.130776470926512222336193233063, −8.260767870769106287090180786639, −7.54015157021754819496049870651, −6.59303828801300462316181832454, −5.75309215520077056146155225593, −4.94478601931509686810841352510, −4.06118293180043275669739473592, −3.71543918689690302085460324654, −2.46103386425163115771496728981, −0.879691661972770754040136248880, 0.879691661972770754040136248880, 2.46103386425163115771496728981, 3.71543918689690302085460324654, 4.06118293180043275669739473592, 4.94478601931509686810841352510, 5.75309215520077056146155225593, 6.59303828801300462316181832454, 7.54015157021754819496049870651, 8.260767870769106287090180786639, 9.130776470926512222336193233063

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