Properties

Label 2-4680-1.1-c1-0-14
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 + 2.40·7-s − 5.38·11-s − 13-s + 7.17·17-s + 7.79·19-s + 2.40·23-s + 25-s − 4.97·29-s − 6.76·31-s − 2.40·35-s + 7.38·37-s − 3.38·41-s − 11.5·43-s + 10.7·47-s − 1.19·49-s − 1.43·53-s + 5.38·55-s − 5.94·59-s − 1.43·61-s + 65-s + 7.58·67-s + 2.61·71-s + 7.02·73-s − 12.9·77-s + 2.61·79-s − 1.94·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.910·7-s − 1.62·11-s − 0.277·13-s + 1.73·17-s + 1.78·19-s + 0.502·23-s + 0.200·25-s − 0.923·29-s − 1.21·31-s − 0.407·35-s + 1.21·37-s − 0.528·41-s − 1.76·43-s + 1.56·47-s − 0.171·49-s − 0.197·53-s + 0.725·55-s − 0.774·59-s − 0.183·61-s + 0.124·65-s + 0.926·67-s + 0.310·71-s + 0.822·73-s − 1.47·77-s + 0.294·79-s − 0.213·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.834714711\)
\(L(\frac12)\) \(\approx\) \(1.834714711\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.40T + 7T^{2} \)
11 \( 1 + 5.38T + 11T^{2} \)
17 \( 1 - 7.17T + 17T^{2} \)
19 \( 1 - 7.79T + 19T^{2} \)
23 \( 1 - 2.40T + 23T^{2} \)
29 \( 1 + 4.97T + 29T^{2} \)
31 \( 1 + 6.76T + 31T^{2} \)
37 \( 1 - 7.38T + 37T^{2} \)
41 \( 1 + 3.38T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 1.43T + 53T^{2} \)
59 \( 1 + 5.94T + 59T^{2} \)
61 \( 1 + 1.43T + 61T^{2} \)
67 \( 1 - 7.58T + 67T^{2} \)
71 \( 1 - 2.61T + 71T^{2} \)
73 \( 1 - 7.02T + 73T^{2} \)
79 \( 1 - 2.61T + 79T^{2} \)
83 \( 1 + 1.94T + 83T^{2} \)
89 \( 1 + 0.618T + 89T^{2} \)
97 \( 1 - 9.22T + 97T^{2} \)
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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

−8.000289408987788265497416592102, −7.62561436972576393989671593018, −7.29601248160645284847637466286, −5.87287347640101518731864392510, −5.18071342372119117624144933659, −4.93086978502295908104881366942, −3.58124676219254280632695343550, −3.03140229178168389599854464624, −1.89243048336532050898243701258, −0.76194580856367103969879017422, 0.76194580856367103969879017422, 1.89243048336532050898243701258, 3.03140229178168389599854464624, 3.58124676219254280632695343550, 4.93086978502295908104881366942, 5.18071342372119117624144933659, 5.87287347640101518731864392510, 7.29601248160645284847637466286, 7.62561436972576393989671593018, 8.000289408987788265497416592102

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