Properties

Label 2-4680-1.1-c1-0-27
Degree $2$
Conductor $4680$
Sign $1$
Analytic cond. $37.3699$
Root an. cond. $6.11309$
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  − 5-s + 1.56·7-s + 5.56·11-s + 13-s + 6.68·17-s + 3.12·19-s + 5.56·23-s + 25-s + 2·29-s − 7.12·31-s − 1.56·35-s + 9.80·37-s − 2.68·41-s − 10.2·43-s − 7.12·47-s − 4.56·49-s − 3.56·53-s − 5.56·55-s + 8·59-s − 10.6·61-s − 65-s − 14.2·67-s − 4.68·71-s + 16.2·73-s + 8.68·77-s + 11.8·79-s + 8·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.590·7-s + 1.67·11-s + 0.277·13-s + 1.62·17-s + 0.716·19-s + 1.15·23-s + 0.200·25-s + 0.371·29-s − 1.27·31-s − 0.263·35-s + 1.61·37-s − 0.419·41-s − 1.56·43-s − 1.03·47-s − 0.651·49-s − 0.489·53-s − 0.749·55-s + 1.04·59-s − 1.36·61-s − 0.124·65-s − 1.74·67-s − 0.555·71-s + 1.90·73-s + 0.989·77-s + 1.32·79-s + 0.878·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4680\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(37.3699\)
Root analytic conductor: \(6.11309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.486360443\)
\(L(\frac12)\) \(\approx\) \(2.486360443\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
13 \( 1 - T \)
good7 \( 1 - 1.56T + 7T^{2} \)
11 \( 1 - 5.56T + 11T^{2} \)
17 \( 1 - 6.68T + 17T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
23 \( 1 - 5.56T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 7.12T + 31T^{2} \)
37 \( 1 - 9.80T + 37T^{2} \)
41 \( 1 + 2.68T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 7.12T + 47T^{2} \)
53 \( 1 + 3.56T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 + 10.6T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 + 4.68T + 71T^{2} \)
73 \( 1 - 16.2T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + 17.8T + 97T^{2} \)
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

−8.156172213138674823885627091360, −7.69738746027565250462421092569, −6.86665299028155737679554142139, −6.21385289254595971035196991562, −5.26350848774820290129026635031, −4.63202992754454959423379687598, −3.60558390242330559636734911977, −3.20266951605659161800882819320, −1.63928801998691322522002884310, −0.985773035544421462851908715762, 0.985773035544421462851908715762, 1.63928801998691322522002884310, 3.20266951605659161800882819320, 3.60558390242330559636734911977, 4.63202992754454959423379687598, 5.26350848774820290129026635031, 6.21385289254595971035196991562, 6.86665299028155737679554142139, 7.69738746027565250462421092569, 8.156172213138674823885627091360

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