Properties

Label 2-7623-1.1-c1-0-244
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  + 2.73·2-s + 5.46·4-s + 3.73·5-s + 7-s + 9.46·8-s + 10.1·10-s + 0.732·13-s + 2.73·14-s + 14.9·16-s − 0.267·17-s − 8.19·19-s + 20.3·20-s − 6.73·23-s + 8.92·25-s + 2·26-s + 5.46·28-s + 4.73·29-s + 0.535·31-s + 21.8·32-s − 0.732·34-s + 3.73·35-s − 2.53·37-s − 22.3·38-s + 35.3·40-s + 4·41-s + 6.46·43-s − 18.3·46-s + ⋯
L(s)  = 1  + 1.93·2-s + 2.73·4-s + 1.66·5-s + 0.377·7-s + 3.34·8-s + 3.22·10-s + 0.203·13-s + 0.730·14-s + 3.73·16-s − 0.0649·17-s − 1.88·19-s + 4.55·20-s − 1.40·23-s + 1.78·25-s + 0.392·26-s + 1.03·28-s + 0.878·29-s + 0.0962·31-s + 3.86·32-s − 0.125·34-s + 0.630·35-s − 0.416·37-s − 3.63·38-s + 5.58·40-s + 0.624·41-s + 0.985·43-s − 2.71·46-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

\(L(1)\) \(\approx\) \(11.17090793\)
\(L(\frac12)\) \(\approx\) \(11.17090793\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 2T^{2} \)
5 \( 1 - 3.73T + 5T^{2} \)
13 \( 1 - 0.732T + 13T^{2} \)
17 \( 1 + 0.267T + 17T^{2} \)
19 \( 1 + 8.19T + 19T^{2} \)
23 \( 1 + 6.73T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 - 0.535T + 31T^{2} \)
37 \( 1 + 2.53T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 6.46T + 43T^{2} \)
47 \( 1 + 1.19T + 47T^{2} \)
53 \( 1 + 9.46T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 2.19T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 - 3.26T + 73T^{2} \)
79 \( 1 - 7.46T + 79T^{2} \)
83 \( 1 - 7.73T + 83T^{2} \)
89 \( 1 + 2.66T + 89T^{2} \)
97 \( 1 - 6.73T + 97T^{2} \)
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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

−7.63736919324340261613472559676, −6.50602127941327054817591244639, −6.34576238036369101376869259730, −5.80486548497530681074414475484, −5.00712215030050945117590091581, −4.47735199066914661561583178891, −3.71158211495319005647780923355, −2.62279026696080044199097281026, −2.15534812477300282918118396614, −1.47879916872234585177897894189, 1.47879916872234585177897894189, 2.15534812477300282918118396614, 2.62279026696080044199097281026, 3.71158211495319005647780923355, 4.47735199066914661561583178891, 5.00712215030050945117590091581, 5.80486548497530681074414475484, 6.34576238036369101376869259730, 6.50602127941327054817591244639, 7.63736919324340261613472559676

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