Properties

Label 2-7623-1.1-c1-0-78
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.36·2-s − 0.129·4-s + 0.636·5-s + 7-s − 2.91·8-s + 0.870·10-s − 3.59·13-s + 1.36·14-s − 3.72·16-s + 1.46·17-s − 5.85·19-s − 0.0824·20-s + 8.56·23-s − 4.59·25-s − 4.91·26-s − 0.129·28-s + 7.73·29-s − 9.44·31-s + 0.731·32-s + 2·34-s + 0.636·35-s + 5.59·37-s − 8.00·38-s − 1.85·40-s + 8.37·41-s + 1.74·43-s + 11.7·46-s + ⋯
L(s)  = 1  + 0.967·2-s − 0.0647·4-s + 0.284·5-s + 0.377·7-s − 1.02·8-s + 0.275·10-s − 0.997·13-s + 0.365·14-s − 0.931·16-s + 0.354·17-s − 1.34·19-s − 0.0184·20-s + 1.78·23-s − 0.918·25-s − 0.964·26-s − 0.0244·28-s + 1.43·29-s − 1.69·31-s + 0.129·32-s + 0.342·34-s + 0.107·35-s + 0.919·37-s − 1.29·38-s − 0.293·40-s + 1.30·41-s + 0.265·43-s + 1.72·46-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

\(L(1)\) \(\approx\) \(2.695045761\)
\(L(\frac12)\) \(\approx\) \(2.695045761\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.36T + 2T^{2} \)
5 \( 1 - 0.636T + 5T^{2} \)
13 \( 1 + 3.59T + 13T^{2} \)
17 \( 1 - 1.46T + 17T^{2} \)
19 \( 1 + 5.85T + 19T^{2} \)
23 \( 1 - 8.56T + 23T^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 + 9.44T + 31T^{2} \)
37 \( 1 - 5.59T + 37T^{2} \)
41 \( 1 - 8.37T + 41T^{2} \)
43 \( 1 - 1.74T + 43T^{2} \)
47 \( 1 + 4.99T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + 1.85T + 67T^{2} \)
71 \( 1 - 9.66T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 4.51T + 79T^{2} \)
83 \( 1 - 3.08T + 83T^{2} \)
89 \( 1 - 4.19T + 89T^{2} \)
97 \( 1 - 7.44T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.82116225243411739036404377428, −7.01842508533187378312603539545, −6.33567660970279767286833926937, −5.59242548594641137501371642051, −5.00061506095793156474986931483, −4.43479820599443244861896516794, −3.69675929417969286816618255226, −2.75870396242534950895593793844, −2.11361236339331091022617459059, −0.69309319429416206093615435197, 0.69309319429416206093615435197, 2.11361236339331091022617459059, 2.75870396242534950895593793844, 3.69675929417969286816618255226, 4.43479820599443244861896516794, 5.00061506095793156474986931483, 5.59242548594641137501371642051, 6.33567660970279767286833926937, 7.01842508533187378312603539545, 7.82116225243411739036404377428

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