Properties

Label 2-1620-1.1-c3-0-18
Degree $2$
Conductor $1620$
Sign $1$
Analytic cond. $95.5830$
Root an. cond. $9.77666$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·5-s + 8.37·7-s + 9.42·11-s + 43.8·13-s − 73.3·17-s + 54.5·19-s + 181.·23-s + 25·25-s − 147.·29-s + 83.2·31-s + 41.8·35-s − 219.·37-s + 220.·41-s + 223.·43-s + 254.·47-s − 272.·49-s − 394.·53-s + 47.1·55-s + 230.·59-s + 899.·61-s + 219.·65-s − 643.·67-s + 132.·71-s + 543.·73-s + 78.9·77-s − 390.·79-s − 434.·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.452·7-s + 0.258·11-s + 0.935·13-s − 1.04·17-s + 0.658·19-s + 1.64·23-s + 0.200·25-s − 0.941·29-s + 0.482·31-s + 0.202·35-s − 0.974·37-s + 0.839·41-s + 0.791·43-s + 0.790·47-s − 0.795·49-s − 1.02·53-s + 0.115·55-s + 0.508·59-s + 1.88·61-s + 0.418·65-s − 1.17·67-s + 0.221·71-s + 0.870·73-s + 0.116·77-s − 0.555·79-s − 0.574·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $1$
Analytic conductor: \(95.5830\)
Root analytic conductor: \(9.77666\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.840443009\)
\(L(\frac12)\) \(\approx\) \(2.840443009\)
\(L(\frac{5}{2})\) 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 \( 1 - 5T \)
good7 \( 1 - 8.37T + 343T^{2} \)
11 \( 1 - 9.42T + 1.33e3T^{2} \)
13 \( 1 - 43.8T + 2.19e3T^{2} \)
17 \( 1 + 73.3T + 4.91e3T^{2} \)
19 \( 1 - 54.5T + 6.85e3T^{2} \)
23 \( 1 - 181.T + 1.21e4T^{2} \)
29 \( 1 + 147.T + 2.43e4T^{2} \)
31 \( 1 - 83.2T + 2.97e4T^{2} \)
37 \( 1 + 219.T + 5.06e4T^{2} \)
41 \( 1 - 220.T + 6.89e4T^{2} \)
43 \( 1 - 223.T + 7.95e4T^{2} \)
47 \( 1 - 254.T + 1.03e5T^{2} \)
53 \( 1 + 394.T + 1.48e5T^{2} \)
59 \( 1 - 230.T + 2.05e5T^{2} \)
61 \( 1 - 899.T + 2.26e5T^{2} \)
67 \( 1 + 643.T + 3.00e5T^{2} \)
71 \( 1 - 132.T + 3.57e5T^{2} \)
73 \( 1 - 543.T + 3.89e5T^{2} \)
79 \( 1 + 390.T + 4.93e5T^{2} \)
83 \( 1 + 434.T + 5.71e5T^{2} \)
89 \( 1 + 218.T + 7.04e5T^{2} \)
97 \( 1 - 1.77e3T + 9.12e5T^{2} \)
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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

−9.023093194580513344298690590663, −8.395847591468564404165095733259, −7.34649393035448434237161709599, −6.64694784429656375407450811644, −5.74671770727637576897352111440, −4.95746075524638350237215219267, −4.00334090685181457495653972027, −2.95830435385935775869479039249, −1.82851942618860751777351020890, −0.849962482988360793579130702457, 0.849962482988360793579130702457, 1.82851942618860751777351020890, 2.95830435385935775869479039249, 4.00334090685181457495653972027, 4.95746075524638350237215219267, 5.74671770727637576897352111440, 6.64694784429656375407450811644, 7.34649393035448434237161709599, 8.395847591468564404165095733259, 9.023093194580513344298690590663

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