Properties

Label 2-4680-1.1-c1-0-13
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 − 1.55·7-s + 2.15·11-s − 13-s − 0.443·17-s − 3.71·19-s − 7.86·23-s + 25-s + 5.71·29-s − 2·31-s − 1.55·35-s + 4.15·37-s + 4.15·41-s + 12.3·43-s + 12.3·47-s − 4.57·49-s + 10.1·53-s + 2.15·55-s − 4·59-s + 8.15·61-s − 65-s + 10.3·67-s − 5.26·71-s − 0.599·73-s − 3.35·77-s − 3.04·79-s − 4·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.588·7-s + 0.649·11-s − 0.277·13-s − 0.107·17-s − 0.851·19-s − 1.64·23-s + 0.200·25-s + 1.06·29-s − 0.359·31-s − 0.263·35-s + 0.683·37-s + 0.648·41-s + 1.87·43-s + 1.79·47-s − 0.654·49-s + 1.39·53-s + 0.290·55-s − 0.520·59-s + 1.04·61-s − 0.124·65-s + 1.25·67-s − 0.625·71-s − 0.0701·73-s − 0.382·77-s − 0.342·79-s − 0.439·83-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

\(L(1)\) \(\approx\) \(1.834161597\)
\(L(\frac12)\) \(\approx\) \(1.834161597\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.55T + 7T^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
17 \( 1 + 0.443T + 17T^{2} \)
19 \( 1 + 3.71T + 19T^{2} \)
23 \( 1 + 7.86T + 23T^{2} \)
29 \( 1 - 5.71T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 4.15T + 37T^{2} \)
41 \( 1 - 4.15T + 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 8.15T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 5.26T + 71T^{2} \)
73 \( 1 + 0.599T + 73T^{2} \)
79 \( 1 + 3.04T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 7.57T + 89T^{2} \)
97 \( 1 - 6.44T + 97T^{2} \)
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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

−8.389521320233405466234643738857, −7.52987900518149832712763830472, −6.76724335423835114074441494006, −6.09683560726035764568914941064, −5.60923403371999044047012363102, −4.35993031903667738737436733543, −3.95442746746174587667913413715, −2.73642018676201282114846998551, −2.05538361295498634417417695337, −0.74121920156900976747120944559, 0.74121920156900976747120944559, 2.05538361295498634417417695337, 2.73642018676201282114846998551, 3.95442746746174587667913413715, 4.35993031903667738737436733543, 5.60923403371999044047012363102, 6.09683560726035764568914941064, 6.76724335423835114074441494006, 7.52987900518149832712763830472, 8.389521320233405466234643738857

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