Properties

Label 2-7488-1.1-c1-0-21
Degree $2$
Conductor $7488$
Sign $1$
Analytic cond. $59.7919$
Root an. cond. $7.73252$
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  − 1.23·5-s + 1.23·7-s − 2·11-s + 13-s − 4.47·17-s + 5.23·19-s − 2.47·23-s − 3.47·25-s + 4.47·29-s − 7.70·31-s − 1.52·35-s − 0.472·37-s − 1.23·41-s + 6.47·43-s + 6.94·47-s − 5.47·49-s + 8.47·53-s + 2.47·55-s − 8.47·59-s + 12.4·61-s − 1.23·65-s + 6.76·67-s + 4.47·71-s + 0.472·73-s − 2.47·77-s + 8.94·79-s − 7.52·83-s + ⋯
L(s)  = 1  − 0.552·5-s + 0.467·7-s − 0.603·11-s + 0.277·13-s − 1.08·17-s + 1.20·19-s − 0.515·23-s − 0.694·25-s + 0.830·29-s − 1.38·31-s − 0.258·35-s − 0.0776·37-s − 0.193·41-s + 0.986·43-s + 1.01·47-s − 0.781·49-s + 1.16·53-s + 0.333·55-s − 1.10·59-s + 1.59·61-s − 0.153·65-s + 0.826·67-s + 0.530·71-s + 0.0552·73-s − 0.281·77-s + 1.00·79-s − 0.826·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7488\)    =    \(2^{6} \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(59.7919\)
Root analytic conductor: \(7.73252\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7488,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.526565659\)
\(L(\frac12)\) \(\approx\) \(1.526565659\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 - T \)
good5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 - 5.23T + 19T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 7.70T + 31T^{2} \)
37 \( 1 + 0.472T + 37T^{2} \)
41 \( 1 + 1.23T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 - 6.94T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 + 8.47T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 - 6.76T + 67T^{2} \)
71 \( 1 - 4.47T + 71T^{2} \)
73 \( 1 - 0.472T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 7.52T + 83T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + 4.47T + 97T^{2} \)
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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

−7.86584936347846120603435777908, −7.32535914285774143562826187211, −6.59452004421354228476703053154, −5.66086797925403449181845586581, −5.13363215901045875263602091978, −4.23948388958377937929377457355, −3.67112932478109513491380766547, −2.66510764636301660112184374793, −1.83819604305933766987372993174, −0.61241993762452961743326099659, 0.61241993762452961743326099659, 1.83819604305933766987372993174, 2.66510764636301660112184374793, 3.67112932478109513491380766547, 4.23948388958377937929377457355, 5.13363215901045875263602091978, 5.66086797925403449181845586581, 6.59452004421354228476703053154, 7.32535914285774143562826187211, 7.86584936347846120603435777908

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