Properties

Label 2-4680-1.1-c1-0-4
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2.56·7-s + 1.43·11-s + 13-s − 5.68·17-s − 5.12·19-s + 1.43·23-s + 25-s + 2·29-s + 1.12·31-s + 2.56·35-s − 10.8·37-s + 9.68·41-s + 6.24·43-s + 1.12·47-s − 0.438·49-s + 0.561·53-s − 1.43·55-s + 8·59-s + 1.68·61-s − 65-s + 2.24·67-s + 7.68·71-s − 0.246·73-s − 3.68·77-s − 8.80·79-s + 8·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.968·7-s + 0.433·11-s + 0.277·13-s − 1.37·17-s − 1.17·19-s + 0.299·23-s + 0.200·25-s + 0.371·29-s + 0.201·31-s + 0.432·35-s − 1.77·37-s + 1.51·41-s + 0.952·43-s + 0.163·47-s − 0.0626·49-s + 0.0771·53-s − 0.193·55-s + 1.04·59-s + 0.215·61-s − 0.124·65-s + 0.274·67-s + 0.912·71-s − 0.0288·73-s − 0.419·77-s − 0.990·79-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

\(L(1)\) \(\approx\) \(1.136192671\)
\(L(\frac12)\) \(\approx\) \(1.136192671\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.56T + 7T^{2} \)
11 \( 1 - 1.43T + 11T^{2} \)
17 \( 1 + 5.68T + 17T^{2} \)
19 \( 1 + 5.12T + 19T^{2} \)
23 \( 1 - 1.43T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 1.12T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 - 9.68T + 41T^{2} \)
43 \( 1 - 6.24T + 43T^{2} \)
47 \( 1 - 1.12T + 47T^{2} \)
53 \( 1 - 0.561T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 1.68T + 61T^{2} \)
67 \( 1 - 2.24T + 67T^{2} \)
71 \( 1 - 7.68T + 71T^{2} \)
73 \( 1 + 0.246T + 73T^{2} \)
79 \( 1 + 8.80T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 - 2.31T + 89T^{2} \)
97 \( 1 - 2.80T + 97T^{2} \)
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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

−8.495051124554728999223797716528, −7.50169714488355958654239591087, −6.63786260201291640589841428263, −6.44612833185801654666308918877, −5.40042644848309558008843104383, −4.35906421335095256086661752033, −3.88273734661303330469144670286, −2.91972495491246424361349917833, −2.02124778414735920230086391383, −0.56947844604877119435785832240, 0.56947844604877119435785832240, 2.02124778414735920230086391383, 2.91972495491246424361349917833, 3.88273734661303330469144670286, 4.35906421335095256086661752033, 5.40042644848309558008843104383, 6.44612833185801654666308918877, 6.63786260201291640589841428263, 7.50169714488355958654239591087, 8.495051124554728999223797716528

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