Properties

Label 2-2151-1.1-c1-0-46
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  + 0.244·2-s − 1.94·4-s + 4.22·5-s + 0.0748·7-s − 0.962·8-s + 1.03·10-s + 3.64·11-s + 6.36·13-s + 0.0182·14-s + 3.64·16-s − 2.35·17-s + 1.06·19-s − 8.19·20-s + 0.890·22-s − 1.33·23-s + 12.8·25-s + 1.55·26-s − 0.145·28-s − 8.25·29-s − 7.90·31-s + 2.81·32-s − 0.575·34-s + 0.316·35-s − 2.87·37-s + 0.260·38-s − 4.06·40-s + 7.12·41-s + ⋯
L(s)  = 1  + 0.172·2-s − 0.970·4-s + 1.88·5-s + 0.0283·7-s − 0.340·8-s + 0.326·10-s + 1.09·11-s + 1.76·13-s + 0.00489·14-s + 0.911·16-s − 0.571·17-s + 0.244·19-s − 1.83·20-s + 0.189·22-s − 0.277·23-s + 2.56·25-s + 0.304·26-s − 0.0274·28-s − 1.53·29-s − 1.41·31-s + 0.497·32-s − 0.0987·34-s + 0.0534·35-s − 0.473·37-s + 0.0423·38-s − 0.643·40-s + 1.11·41-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

\(L(1)\) \(\approx\) \(2.523649146\)
\(L(\frac12)\) \(\approx\) \(2.523649146\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.244T + 2T^{2} \)
5 \( 1 - 4.22T + 5T^{2} \)
7 \( 1 - 0.0748T + 7T^{2} \)
11 \( 1 - 3.64T + 11T^{2} \)
13 \( 1 - 6.36T + 13T^{2} \)
17 \( 1 + 2.35T + 17T^{2} \)
19 \( 1 - 1.06T + 19T^{2} \)
23 \( 1 + 1.33T + 23T^{2} \)
29 \( 1 + 8.25T + 29T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 + 2.87T + 37T^{2} \)
41 \( 1 - 7.12T + 41T^{2} \)
43 \( 1 - 6.29T + 43T^{2} \)
47 \( 1 - 6.74T + 47T^{2} \)
53 \( 1 - 9.74T + 53T^{2} \)
59 \( 1 + 4.34T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 4.26T + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 - 6.66T + 73T^{2} \)
79 \( 1 - 17.5T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 - 9.62T + 97T^{2} \)
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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

−9.126600722652722613539920993587, −8.809653440630233323516319839324, −7.50961429319168316743500128256, −6.29539296968822784818490977530, −5.97507443174375634158965002325, −5.27055790141603007897746954357, −4.16314597960089564622563614150, −3.42448289583033375062558084773, −1.98824097582349147039884611485, −1.14735846984620016865425758151, 1.14735846984620016865425758151, 1.98824097582349147039884611485, 3.42448289583033375062558084773, 4.16314597960089564622563614150, 5.27055790141603007897746954357, 5.97507443174375634158965002325, 6.29539296968822784818490977530, 7.50961429319168316743500128256, 8.809653440630233323516319839324, 9.126600722652722613539920993587

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