Properties

Label 2-2671-1.1-c1-0-125
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  − 1.51·2-s + 3.03·3-s + 0.307·4-s + 3.69·5-s − 4.60·6-s − 2.45·7-s + 2.57·8-s + 6.18·9-s − 5.60·10-s + 1.97·11-s + 0.933·12-s + 0.285·13-s + 3.72·14-s + 11.1·15-s − 4.52·16-s + 2.89·17-s − 9.39·18-s + 8.63·19-s + 1.13·20-s − 7.43·21-s − 3.00·22-s + 1.21·23-s + 7.79·24-s + 8.61·25-s − 0.434·26-s + 9.65·27-s − 0.755·28-s + ⋯
L(s)  = 1  − 1.07·2-s + 1.74·3-s + 0.153·4-s + 1.65·5-s − 1.87·6-s − 0.927·7-s + 0.908·8-s + 2.06·9-s − 1.77·10-s + 0.595·11-s + 0.269·12-s + 0.0792·13-s + 0.996·14-s + 2.88·15-s − 1.13·16-s + 0.702·17-s − 2.21·18-s + 1.98·19-s + 0.254·20-s − 1.62·21-s − 0.639·22-s + 0.253·23-s + 1.59·24-s + 1.72·25-s − 0.0851·26-s + 1.85·27-s − 0.142·28-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

\(L(1)\) \(\approx\) \(2.613137949\)
\(L(\frac12)\) \(\approx\) \(2.613137949\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.51T + 2T^{2} \)
3 \( 1 - 3.03T + 3T^{2} \)
5 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 + 2.45T + 7T^{2} \)
11 \( 1 - 1.97T + 11T^{2} \)
13 \( 1 - 0.285T + 13T^{2} \)
17 \( 1 - 2.89T + 17T^{2} \)
19 \( 1 - 8.63T + 19T^{2} \)
23 \( 1 - 1.21T + 23T^{2} \)
29 \( 1 + 2.02T + 29T^{2} \)
31 \( 1 + 9.19T + 31T^{2} \)
37 \( 1 - 3.26T + 37T^{2} \)
41 \( 1 - 2.48T + 41T^{2} \)
43 \( 1 + 8.60T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 - 0.882T + 53T^{2} \)
59 \( 1 + 0.471T + 59T^{2} \)
61 \( 1 - 2.66T + 61T^{2} \)
67 \( 1 + 2.11T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + 3.95T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + 3.52T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−9.104381730467014028552557448718, −8.390574809560884295137080565883, −7.46973833229955721364231108436, −6.98930431678973290102760486476, −5.92322037132140657751425184230, −4.95152295297087275409544654990, −3.58950194617927337743581241877, −3.01484576563288553437317781550, −1.88735975834609826280164416126, −1.26592722536952746867650097791, 1.26592722536952746867650097791, 1.88735975834609826280164416126, 3.01484576563288553437317781550, 3.58950194617927337743581241877, 4.95152295297087275409544654990, 5.92322037132140657751425184230, 6.98930431678973290102760486476, 7.46973833229955721364231108436, 8.390574809560884295137080565883, 9.104381730467014028552557448718

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