Properties

Label 2-1620-1.1-c3-0-10
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 + 27.5·7-s − 40.6·11-s − 67.8·13-s − 91.1·17-s + 37.9·19-s − 57.7·23-s + 25·25-s + 49.8·29-s + 226.·31-s − 137.·35-s − 144.·37-s + 348.·41-s + 455.·43-s + 179.·47-s + 418.·49-s − 300.·53-s + 203.·55-s − 439.·59-s − 191.·61-s + 339.·65-s + 462.·67-s + 671.·71-s − 315.·73-s − 1.12e3·77-s + 1.00e3·79-s + 1.20e3·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.48·7-s − 1.11·11-s − 1.44·13-s − 1.29·17-s + 0.457·19-s − 0.523·23-s + 0.200·25-s + 0.319·29-s + 1.30·31-s − 0.666·35-s − 0.640·37-s + 1.32·41-s + 1.61·43-s + 0.556·47-s + 1.21·49-s − 0.779·53-s + 0.498·55-s − 0.970·59-s − 0.402·61-s + 0.647·65-s + 0.843·67-s + 1.12·71-s − 0.506·73-s − 1.65·77-s + 1.43·79-s + 1.59·83-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

\(L(2)\) \(\approx\) \(1.685571772\)
\(L(\frac12)\) \(\approx\) \(1.685571772\)
\(L(\frac{5}{2})\) 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 \( 1 - 27.5T + 343T^{2} \)
11 \( 1 + 40.6T + 1.33e3T^{2} \)
13 \( 1 + 67.8T + 2.19e3T^{2} \)
17 \( 1 + 91.1T + 4.91e3T^{2} \)
19 \( 1 - 37.9T + 6.85e3T^{2} \)
23 \( 1 + 57.7T + 1.21e4T^{2} \)
29 \( 1 - 49.8T + 2.43e4T^{2} \)
31 \( 1 - 226.T + 2.97e4T^{2} \)
37 \( 1 + 144.T + 5.06e4T^{2} \)
41 \( 1 - 348.T + 6.89e4T^{2} \)
43 \( 1 - 455.T + 7.95e4T^{2} \)
47 \( 1 - 179.T + 1.03e5T^{2} \)
53 \( 1 + 300.T + 1.48e5T^{2} \)
59 \( 1 + 439.T + 2.05e5T^{2} \)
61 \( 1 + 191.T + 2.26e5T^{2} \)
67 \( 1 - 462.T + 3.00e5T^{2} \)
71 \( 1 - 671.T + 3.57e5T^{2} \)
73 \( 1 + 315.T + 3.89e5T^{2} \)
79 \( 1 - 1.00e3T + 4.93e5T^{2} \)
83 \( 1 - 1.20e3T + 5.71e5T^{2} \)
89 \( 1 + 981.T + 7.04e5T^{2} \)
97 \( 1 + 1.84e3T + 9.12e5T^{2} \)
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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

−8.942963777612603846325140185925, −7.88035972290058235255925239115, −7.81501370757524515042672825284, −6.78341517366664482641487384202, −5.55325774413622143849596446167, −4.76196750944738910989426328676, −4.32532608457883809219490899534, −2.74581869551032436106933433282, −2.06440495359369255977754401300, −0.60679390505036620557956778631, 0.60679390505036620557956778631, 2.06440495359369255977754401300, 2.74581869551032436106933433282, 4.32532608457883809219490899534, 4.76196750944738910989426328676, 5.55325774413622143849596446167, 6.78341517366664482641487384202, 7.81501370757524515042672825284, 7.88035972290058235255925239115, 8.942963777612603846325140185925

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