Properties

Label 2-7623-1.1-c1-0-92
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  − 1.61·2-s + 0.618·4-s − 0.618·5-s + 7-s + 2.23·8-s + 1.00·10-s + 0.236·13-s − 1.61·14-s − 4.85·16-s + 5.47·17-s + 2.47·19-s − 0.381·20-s + 7.70·23-s − 4.61·25-s − 0.381·26-s + 0.618·28-s + 4.23·29-s + 3·31-s + 3.38·32-s − 8.85·34-s − 0.618·35-s + 6·37-s − 4.00·38-s − 1.38·40-s − 3.14·41-s + 2.52·43-s − 12.4·46-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.309·4-s − 0.276·5-s + 0.377·7-s + 0.790·8-s + 0.316·10-s + 0.0654·13-s − 0.432·14-s − 1.21·16-s + 1.32·17-s + 0.567·19-s − 0.0854·20-s + 1.60·23-s − 0.923·25-s − 0.0749·26-s + 0.116·28-s + 0.786·29-s + 0.538·31-s + 0.597·32-s − 1.51·34-s − 0.104·35-s + 0.986·37-s − 0.648·38-s − 0.218·40-s − 0.491·41-s + 0.385·43-s − 1.83·46-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

\(L(1)\) \(\approx\) \(1.186636404\)
\(L(\frac12)\) \(\approx\) \(1.186636404\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.61T + 2T^{2} \)
5 \( 1 + 0.618T + 5T^{2} \)
13 \( 1 - 0.236T + 13T^{2} \)
17 \( 1 - 5.47T + 17T^{2} \)
19 \( 1 - 2.47T + 19T^{2} \)
23 \( 1 - 7.70T + 23T^{2} \)
29 \( 1 - 4.23T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 3.14T + 41T^{2} \)
43 \( 1 - 2.52T + 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + 7.56T + 53T^{2} \)
59 \( 1 - 2.38T + 59T^{2} \)
61 \( 1 + 9.94T + 61T^{2} \)
67 \( 1 - 8.76T + 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 + 9.61T + 73T^{2} \)
79 \( 1 + 5.09T + 79T^{2} \)
83 \( 1 + 0.472T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 18.7T + 97T^{2} \)
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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

−7.88218300085692463824941043998, −7.51979196432825310572940976308, −6.78684231885080688254529925261, −5.81490278903582699590296907835, −5.04573542166193913389101179879, −4.37634709173101919282573637263, −3.44207542904080822939567517731, −2.53572889170397634236288042937, −1.34945395420141126998493838498, −0.75048958752610155151681797335, 0.75048958752610155151681797335, 1.34945395420141126998493838498, 2.53572889170397634236288042937, 3.44207542904080822939567517731, 4.37634709173101919282573637263, 5.04573542166193913389101179879, 5.81490278903582699590296907835, 6.78684231885080688254529925261, 7.51979196432825310572940976308, 7.88218300085692463824941043998

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