Properties

Label 2-2671-1.1-c1-0-112
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  − 2.46·2-s + 3.35·3-s + 4.05·4-s − 0.322·5-s − 8.24·6-s + 3.62·7-s − 5.05·8-s + 8.23·9-s + 0.793·10-s + 4.00·11-s + 13.5·12-s − 3.84·13-s − 8.92·14-s − 1.08·15-s + 4.33·16-s − 0.699·17-s − 20.2·18-s − 4.67·19-s − 1.30·20-s + 12.1·21-s − 9.85·22-s − 2.04·23-s − 16.9·24-s − 4.89·25-s + 9.45·26-s + 17.5·27-s + 14.7·28-s + ⋯
L(s)  = 1  − 1.73·2-s + 1.93·3-s + 2.02·4-s − 0.144·5-s − 3.36·6-s + 1.37·7-s − 1.78·8-s + 2.74·9-s + 0.251·10-s + 1.20·11-s + 3.92·12-s − 1.06·13-s − 2.38·14-s − 0.279·15-s + 1.08·16-s − 0.169·17-s − 4.77·18-s − 1.07·19-s − 0.292·20-s + 2.65·21-s − 2.10·22-s − 0.426·23-s − 3.45·24-s − 0.979·25-s + 1.85·26-s + 3.37·27-s + 2.78·28-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

\(L(1)\) \(\approx\) \(2.045593584\)
\(L(\frac12)\) \(\approx\) \(2.045593584\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.46T + 2T^{2} \)
3 \( 1 - 3.35T + 3T^{2} \)
5 \( 1 + 0.322T + 5T^{2} \)
7 \( 1 - 3.62T + 7T^{2} \)
11 \( 1 - 4.00T + 11T^{2} \)
13 \( 1 + 3.84T + 13T^{2} \)
17 \( 1 + 0.699T + 17T^{2} \)
19 \( 1 + 4.67T + 19T^{2} \)
23 \( 1 + 2.04T + 23T^{2} \)
29 \( 1 - 4.68T + 29T^{2} \)
31 \( 1 + 8.16T + 31T^{2} \)
37 \( 1 - 9.85T + 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 - 7.42T + 43T^{2} \)
47 \( 1 + 5.54T + 47T^{2} \)
53 \( 1 - 4.57T + 53T^{2} \)
59 \( 1 - 5.16T + 59T^{2} \)
61 \( 1 + 0.171T + 61T^{2} \)
67 \( 1 - 1.72T + 67T^{2} \)
71 \( 1 + 9.41T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 + 7.27T + 79T^{2} \)
83 \( 1 + 5.50T + 83T^{2} \)
89 \( 1 - 7.89T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−8.752314446407184981133056125462, −8.285611851078170692149456096350, −7.57932223799200682179412785758, −7.33649788593189596979632961845, −6.26944952999172744993290476633, −4.51984953635856501650140447524, −3.94395395862037878012579695213, −2.46822131585005616279179995574, −2.07414647680341993468454060669, −1.15573446387713801950363262962, 1.15573446387713801950363262962, 2.07414647680341993468454060669, 2.46822131585005616279179995574, 3.94395395862037878012579695213, 4.51984953635856501650140447524, 6.26944952999172744993290476633, 7.33649788593189596979632961845, 7.57932223799200682179412785758, 8.285611851078170692149456096350, 8.752314446407184981133056125462

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