Properties

Label 2-6762-1.1-c1-0-76
Degree $2$
Conductor $6762$
Sign $1$
Analytic cond. $53.9948$
Root an. cond. $7.34811$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 3·5-s − 6-s + 8-s + 9-s + 3·10-s + 2·11-s − 12-s + 5·13-s − 3·15-s + 16-s + 4·17-s + 18-s − 6·19-s + 3·20-s + 2·22-s + 23-s − 24-s + 4·25-s + 5·26-s − 27-s + 29-s − 3·30-s + 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.603·11-s − 0.288·12-s + 1.38·13-s − 0.774·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.37·19-s + 0.670·20-s + 0.426·22-s + 0.208·23-s − 0.204·24-s + 4/5·25-s + 0.980·26-s − 0.192·27-s + 0.185·29-s − 0.547·30-s + 0.718·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6762\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(53.9948\)
Root analytic conductor: \(7.34811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{6762} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6762,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.203054212\)
\(L(\frac12)\) \(\approx\) \(4.203054212\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
23 \( 1 - T \)
good5 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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

−7.918617881851957863498926850459, −6.86968250552726582021652091341, −6.28137344671932013165462585347, −5.96153762451085121171496200820, −5.30976521983415597175653776546, −4.42324800736466071285079030366, −3.73650977170265112461812457567, −2.74464353211888054441904822657, −1.77168877155839281306683984426, −1.07167247769192420833946347945, 1.07167247769192420833946347945, 1.77168877155839281306683984426, 2.74464353211888054441904822657, 3.73650977170265112461812457567, 4.42324800736466071285079030366, 5.30976521983415597175653776546, 5.96153762451085121171496200820, 6.28137344671932013165462585347, 6.86968250552726582021652091341, 7.918617881851957863498926850459

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