Properties

Label 2-2671-1.1-c1-0-131
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  − 2.39·2-s + 2.49·3-s + 3.74·4-s + 2.75·5-s − 5.97·6-s + 1.87·7-s − 4.18·8-s + 3.21·9-s − 6.61·10-s + 3.35·11-s + 9.33·12-s + 0.693·13-s − 4.49·14-s + 6.88·15-s + 2.53·16-s + 4.26·17-s − 7.71·18-s + 2.53·19-s + 10.3·20-s + 4.68·21-s − 8.04·22-s − 1.84·23-s − 10.4·24-s + 2.61·25-s − 1.66·26-s + 0.548·27-s + 7.02·28-s + ⋯
L(s)  = 1  − 1.69·2-s + 1.43·3-s + 1.87·4-s + 1.23·5-s − 2.44·6-s + 0.709·7-s − 1.47·8-s + 1.07·9-s − 2.09·10-s + 1.01·11-s + 2.69·12-s + 0.192·13-s − 1.20·14-s + 1.77·15-s + 0.632·16-s + 1.03·17-s − 1.81·18-s + 0.581·19-s + 2.31·20-s + 1.02·21-s − 1.71·22-s − 0.384·23-s − 2.12·24-s + 0.522·25-s − 0.325·26-s + 0.105·27-s + 1.32·28-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

\(L(1)\) \(\approx\) \(2.162067939\)
\(L(\frac12)\) \(\approx\) \(2.162067939\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.39T + 2T^{2} \)
3 \( 1 - 2.49T + 3T^{2} \)
5 \( 1 - 2.75T + 5T^{2} \)
7 \( 1 - 1.87T + 7T^{2} \)
11 \( 1 - 3.35T + 11T^{2} \)
13 \( 1 - 0.693T + 13T^{2} \)
17 \( 1 - 4.26T + 17T^{2} \)
19 \( 1 - 2.53T + 19T^{2} \)
23 \( 1 + 1.84T + 23T^{2} \)
29 \( 1 - 0.183T + 29T^{2} \)
31 \( 1 - 8.57T + 31T^{2} \)
37 \( 1 + 5.83T + 37T^{2} \)
41 \( 1 + 11.5T + 41T^{2} \)
43 \( 1 - 0.323T + 43T^{2} \)
47 \( 1 + 6.39T + 47T^{2} \)
53 \( 1 - 6.34T + 53T^{2} \)
59 \( 1 + 8.13T + 59T^{2} \)
61 \( 1 - 1.21T + 61T^{2} \)
67 \( 1 - 9.07T + 67T^{2} \)
71 \( 1 - 8.81T + 71T^{2} \)
73 \( 1 + 15.7T + 73T^{2} \)
79 \( 1 - 1.32T + 79T^{2} \)
83 \( 1 - 5.26T + 83T^{2} \)
89 \( 1 + 0.368T + 89T^{2} \)
97 \( 1 + 9.66T + 97T^{2} \)
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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

−8.767287973024019010888982031859, −8.362039972936804545390604278215, −7.74778641718787343354838870935, −6.89499046208758767206336372300, −6.15516217014856062745904568287, −5.02132580079844485192221855372, −3.63176370771838033983647952465, −2.69159951903203358015617657220, −1.74313666500906568793081112560, −1.31926287827250957463480591447, 1.31926287827250957463480591447, 1.74313666500906568793081112560, 2.69159951903203358015617657220, 3.63176370771838033983647952465, 5.02132580079844485192221855372, 6.15516217014856062745904568287, 6.89499046208758767206336372300, 7.74778641718787343354838870935, 8.362039972936804545390604278215, 8.767287973024019010888982031859

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