↑

Properties

Label 2-7623-1.1-c1-0-223
Degree $2$
Conductor $7623$
Sign $-1$
Analytic cond. $60.8699$
Root an. cond. $7.80192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 3·5-s − 7-s + 4·13-s + 4·16-s − 3·17-s − 2·19-s − 6·20-s + 4·25-s + 2·28-s − 6·29-s + 2·31-s − 3·35-s − 10·37-s + 6·41-s − 11·43-s + 3·47-s + 49-s − 8·52-s − 12·53-s + 3·59-s − 8·61-s − 8·64-s + 12·65-s + 5·67-s + 6·68-s + 12·71-s + ⋯
L(s)  = 1  − 4-s + 1.34·5-s − 0.377·7-s + 1.10·13-s + 16-s − 0.727·17-s − 0.458·19-s − 1.34·20-s + 4/5·25-s + 0.377·28-s − 1.11·29-s + 0.359·31-s − 0.507·35-s − 1.64·37-s + 0.937·41-s − 1.67·43-s + 0.437·47-s + 1/7·49-s − 1.10·52-s − 1.64·53-s + 0.390·59-s − 1.02·61-s − 64-s + 1.48·65-s + 0.610·67-s + 0.727·68-s + 1.42·71-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7623,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 15 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−7.57418343289284618677806980173, −6.55940367910692825238465691228, −6.14129429934855910937378306050, −5.44736663243645956885616895821, −4.80543469905603584152591547401, −3.90505472796203169010606472173, −3.24068751357251189785471323455, −2.10298053732574542896093056018, −1.33894202221770869479377057432, 0, 1.33894202221770869479377057432, 2.10298053732574542896093056018, 3.24068751357251189785471323455, 3.90505472796203169010606472173, 4.80543469905603584152591547401, 5.44736663243645956885616895821, 6.14129429934855910937378306050, 6.55940367910692825238465691228, 7.57418343289284618677806980173

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