Properties

Label 2-2671-1.1-c1-0-135
Degree $2$
Conductor $2671$
Sign $1$
Analytic cond. $21.3280$
Root an. cond. $4.61822$
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  + 1.67·2-s + 2.07·3-s + 0.820·4-s + 1.54·5-s + 3.48·6-s + 0.0416·7-s − 1.98·8-s + 1.29·9-s + 2.59·10-s − 0.546·11-s + 1.70·12-s + 4.01·13-s + 0.0699·14-s + 3.20·15-s − 4.96·16-s + 2.09·17-s + 2.17·18-s + 7.55·19-s + 1.26·20-s + 0.0863·21-s − 0.917·22-s + 7.78·23-s − 4.10·24-s − 2.60·25-s + 6.74·26-s − 3.53·27-s + 0.0341·28-s + ⋯
L(s)  = 1  + 1.18·2-s + 1.19·3-s + 0.410·4-s + 0.691·5-s + 1.42·6-s + 0.0157·7-s − 0.700·8-s + 0.432·9-s + 0.821·10-s − 0.164·11-s + 0.490·12-s + 1.11·13-s + 0.0186·14-s + 0.827·15-s − 1.24·16-s + 0.508·17-s + 0.513·18-s + 1.73·19-s + 0.283·20-s + 0.0188·21-s − 0.195·22-s + 1.62·23-s − 0.838·24-s − 0.521·25-s + 1.32·26-s − 0.679·27-s + 0.00645·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2671\)
Sign: $1$
Analytic conductor: \(21.3280\)
Root analytic conductor: \(4.61822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2671,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.595316185\)
\(L(\frac12)\) \(\approx\) \(5.595316185\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2671 \( 1 - T \)
good2 \( 1 - 1.67T + 2T^{2} \)
3 \( 1 - 2.07T + 3T^{2} \)
5 \( 1 - 1.54T + 5T^{2} \)
7 \( 1 - 0.0416T + 7T^{2} \)
11 \( 1 + 0.546T + 11T^{2} \)
13 \( 1 - 4.01T + 13T^{2} \)
17 \( 1 - 2.09T + 17T^{2} \)
19 \( 1 - 7.55T + 19T^{2} \)
23 \( 1 - 7.78T + 23T^{2} \)
29 \( 1 + 5.67T + 29T^{2} \)
31 \( 1 - 4.37T + 31T^{2} \)
37 \( 1 - 3.83T + 37T^{2} \)
41 \( 1 + 5.06T + 41T^{2} \)
43 \( 1 - 5.71T + 43T^{2} \)
47 \( 1 - 2.07T + 47T^{2} \)
53 \( 1 + 6.18T + 53T^{2} \)
59 \( 1 - 4.20T + 59T^{2} \)
61 \( 1 - 0.0854T + 61T^{2} \)
67 \( 1 + 4.05T + 67T^{2} \)
71 \( 1 - 6.98T + 71T^{2} \)
73 \( 1 + 7.44T + 73T^{2} \)
79 \( 1 + 14.5T + 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + 1.26T + 89T^{2} \)
97 \( 1 + 3.54T + 97T^{2} \)
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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

−9.042077251108210560915473929609, −8.082238662916731678079771693016, −7.35186849972162643255390921950, −6.29042341031621853083959948896, −5.61103564202452745204937200031, −4.96984658103352120688746763058, −3.81867673868707092545783160962, −3.24075261965139839682920603290, −2.60548308214549507173441092210, −1.34302521965891774808465133442, 1.34302521965891774808465133442, 2.60548308214549507173441092210, 3.24075261965139839682920603290, 3.81867673868707092545783160962, 4.96984658103352120688746763058, 5.61103564202452745204937200031, 6.29042341031621853083959948896, 7.35186849972162643255390921950, 8.082238662916731678079771693016, 9.042077251108210560915473929609

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