Properties

Label 2-4560-1.1-c1-0-9
Degree $2$
Conductor $4560$
Sign $1$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 4·7-s + 9-s + 4·11-s + 2·13-s − 15-s − 6·17-s − 19-s + 4·21-s − 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s − 4·33-s − 4·35-s + 10·37-s − 2·39-s + 2·41-s + 45-s + 4·47-s + 9·49-s + 6·51-s + 2·53-s + 4·55-s + 57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.229·19-s + 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.676·35-s + 1.64·37-s − 0.320·39-s + 0.312·41-s + 0.149·45-s + 0.583·47-s + 9/7·49-s + 0.840·51-s + 0.274·53-s + 0.539·55-s + 0.132·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.247622961\)
\(L(\frac12)\) \(\approx\) \(1.247622961\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - T \)
19 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.597868213785013283294424214525, −7.29727168098159772066066874094, −6.67346952173222818736818531256, −6.17701843493987983428915779478, −5.73811103545833013445606483311, −4.40522243267350251888189588326, −3.94543152310830867263732506887, −2.91353064096016451096410866873, −1.88609460132414588878093443132, −0.63292907005131187915961561972, 0.63292907005131187915961561972, 1.88609460132414588878093443132, 2.91353064096016451096410866873, 3.94543152310830867263732506887, 4.40522243267350251888189588326, 5.73811103545833013445606483311, 6.17701843493987983428915779478, 6.67346952173222818736818531256, 7.29727168098159772066066874094, 8.597868213785013283294424214525

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