Properties

Label 2-7623-1.1-c1-0-14
Degree $2$
Conductor $7623$
Sign $1$
Analytic cond. $60.8699$
Root an. cond. $7.80192$
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  − 0.473·2-s − 1.77·4-s − 3.75·5-s − 7-s + 1.78·8-s + 1.78·10-s + 2.70·13-s + 0.473·14-s + 2.70·16-s − 2.48·17-s − 1.74·19-s + 6.67·20-s − 3.79·23-s + 9.12·25-s − 1.28·26-s + 1.77·28-s + 5.80·29-s + 2.24·31-s − 4.85·32-s + 1.17·34-s + 3.75·35-s − 7.65·37-s + 0.826·38-s − 6.72·40-s + 4.18·41-s + 7.75·43-s + 1.79·46-s + ⋯
L(s)  = 1  − 0.334·2-s − 0.887·4-s − 1.68·5-s − 0.377·7-s + 0.632·8-s + 0.563·10-s + 0.750·13-s + 0.126·14-s + 0.675·16-s − 0.602·17-s − 0.400·19-s + 1.49·20-s − 0.792·23-s + 1.82·25-s − 0.251·26-s + 0.335·28-s + 1.07·29-s + 0.403·31-s − 0.858·32-s + 0.201·34-s + 0.635·35-s − 1.25·37-s + 0.134·38-s − 1.06·40-s + 0.653·41-s + 1.18·43-s + 0.265·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7623\)    =    \(3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(60.8699\)
Root analytic conductor: \(7.80192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7623,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3620665469\)
\(L(\frac12)\) \(\approx\) \(0.3620665469\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 0.473T + 2T^{2} \)
5 \( 1 + 3.75T + 5T^{2} \)
13 \( 1 - 2.70T + 13T^{2} \)
17 \( 1 + 2.48T + 17T^{2} \)
19 \( 1 + 1.74T + 19T^{2} \)
23 \( 1 + 3.79T + 23T^{2} \)
29 \( 1 - 5.80T + 29T^{2} \)
31 \( 1 - 2.24T + 31T^{2} \)
37 \( 1 + 7.65T + 37T^{2} \)
41 \( 1 - 4.18T + 41T^{2} \)
43 \( 1 - 7.75T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 + 8.83T + 53T^{2} \)
59 \( 1 + 4.54T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 + 7.75T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 7.05T + 79T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 - 1.69T + 89T^{2} \)
97 \( 1 - 6.99T + 97T^{2} \)
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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

−8.114066689423162206261271678330, −7.37795847215074781279993094294, −6.60514682667635848107793923858, −5.85157126385971822856153100961, −4.64373845234282289847013390329, −4.40897107732851822060691057819, −3.62227175241455372309276604471, −2.98858466634040221393819382683, −1.46885090353247755508536486113, −0.33586473709459875090234456087, 0.33586473709459875090234456087, 1.46885090353247755508536486113, 2.98858466634040221393819382683, 3.62227175241455372309276604471, 4.40897107732851822060691057819, 4.64373845234282289847013390329, 5.85157126385971822856153100961, 6.60514682667635848107793923858, 7.37795847215074781279993094294, 8.114066689423162206261271678330

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