Properties

Label 2-1620-1.1-c3-0-1
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 − 29.9·7-s − 56.7·11-s + 30.3·13-s − 99.8·17-s − 81.8·19-s − 137.·23-s + 25·25-s + 49.0·29-s + 3.50·31-s + 149.·35-s − 289.·37-s + 199.·41-s − 516.·43-s + 109.·47-s + 552.·49-s − 283.·53-s + 283.·55-s + 185.·59-s − 235.·61-s − 151.·65-s − 341.·67-s − 1.13e3·71-s + 1.11e3·73-s + 1.69e3·77-s − 200.·79-s − 205.·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.61·7-s − 1.55·11-s + 0.648·13-s − 1.42·17-s − 0.988·19-s − 1.24·23-s + 0.200·25-s + 0.314·29-s + 0.0203·31-s + 0.722·35-s − 1.28·37-s + 0.761·41-s − 1.83·43-s + 0.340·47-s + 1.61·49-s − 0.734·53-s + 0.696·55-s + 0.409·59-s − 0.495·61-s − 0.289·65-s − 0.622·67-s − 1.89·71-s + 1.78·73-s + 2.51·77-s − 0.285·79-s − 0.271·83-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

\(L(2)\) \(\approx\) \(0.1565050829\)
\(L(\frac12)\) \(\approx\) \(0.1565050829\)
\(L(\frac{5}{2})\) 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 \( 1 + 29.9T + 343T^{2} \)
11 \( 1 + 56.7T + 1.33e3T^{2} \)
13 \( 1 - 30.3T + 2.19e3T^{2} \)
17 \( 1 + 99.8T + 4.91e3T^{2} \)
19 \( 1 + 81.8T + 6.85e3T^{2} \)
23 \( 1 + 137.T + 1.21e4T^{2} \)
29 \( 1 - 49.0T + 2.43e4T^{2} \)
31 \( 1 - 3.50T + 2.97e4T^{2} \)
37 \( 1 + 289.T + 5.06e4T^{2} \)
41 \( 1 - 199.T + 6.89e4T^{2} \)
43 \( 1 + 516.T + 7.95e4T^{2} \)
47 \( 1 - 109.T + 1.03e5T^{2} \)
53 \( 1 + 283.T + 1.48e5T^{2} \)
59 \( 1 - 185.T + 2.05e5T^{2} \)
61 \( 1 + 235.T + 2.26e5T^{2} \)
67 \( 1 + 341.T + 3.00e5T^{2} \)
71 \( 1 + 1.13e3T + 3.57e5T^{2} \)
73 \( 1 - 1.11e3T + 3.89e5T^{2} \)
79 \( 1 + 200.T + 4.93e5T^{2} \)
83 \( 1 + 205.T + 5.71e5T^{2} \)
89 \( 1 + 350.T + 7.04e5T^{2} \)
97 \( 1 - 923.T + 9.12e5T^{2} \)
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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

−8.890981278349851774338571385524, −8.362306126386360698817546207776, −7.39188895209236413369437681333, −6.52245087306904214347838057913, −5.99906850419420630500811954812, −4.82674720312887052551552862003, −3.87748834217502946943797421438, −3.03036276122505137001619475961, −2.10303245303226050743036577134, −0.17505438123884286030761215195, 0.17505438123884286030761215195, 2.10303245303226050743036577134, 3.03036276122505137001619475961, 3.87748834217502946943797421438, 4.82674720312887052551552862003, 5.99906850419420630500811954812, 6.52245087306904214347838057913, 7.39188895209236413369437681333, 8.362306126386360698817546207776, 8.890981278349851774338571385524

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