Properties

Label 2-4560-1.1-c1-0-2
Degree $2$
Conductor $4560$
Sign $1$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
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  − 3-s − 5-s − 2.91·7-s + 9-s − 3.43·11-s + 4.91·13-s + 15-s − 4.35·17-s + 19-s + 2.91·21-s − 6.35·23-s + 25-s − 27-s − 7.27·29-s + 2.35·31-s + 3.43·33-s + 2.91·35-s + 4.91·37-s − 4.91·39-s + 1.43·41-s − 6.91·43-s − 45-s + 6.35·47-s + 1.51·49-s + 4.35·51-s − 4.35·53-s + 3.43·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 1.10·7-s + 0.333·9-s − 1.03·11-s + 1.36·13-s + 0.258·15-s − 1.05·17-s + 0.229·19-s + 0.636·21-s − 1.32·23-s + 0.200·25-s − 0.192·27-s − 1.35·29-s + 0.422·31-s + 0.598·33-s + 0.493·35-s + 0.808·37-s − 0.787·39-s + 0.224·41-s − 1.05·43-s − 0.149·45-s + 0.926·47-s + 0.216·49-s + 0.609·51-s − 0.598·53-s + 0.463·55-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: no
Arithmetic: yes
Character: $\chi_{4560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6422514274\)
\(L(\frac12)\) \(\approx\) \(0.6422514274\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.91T + 7T^{2} \)
11 \( 1 + 3.43T + 11T^{2} \)
13 \( 1 - 4.91T + 13T^{2} \)
17 \( 1 + 4.35T + 17T^{2} \)
23 \( 1 + 6.35T + 23T^{2} \)
29 \( 1 + 7.27T + 29T^{2} \)
31 \( 1 - 2.35T + 31T^{2} \)
37 \( 1 - 4.91T + 37T^{2} \)
41 \( 1 - 1.43T + 41T^{2} \)
43 \( 1 + 6.91T + 43T^{2} \)
47 \( 1 - 6.35T + 47T^{2} \)
53 \( 1 + 4.35T + 53T^{2} \)
59 \( 1 + 9.83T + 59T^{2} \)
61 \( 1 + 6.19T + 61T^{2} \)
67 \( 1 - 8.70T + 67T^{2} \)
71 \( 1 + 6.87T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 8.70T + 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + 4.40T + 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−8.194349244145207073905482537800, −7.65573518970770487815204685208, −6.69937079791830339775835528432, −6.16938542931036424871859811240, −5.56715452265414334711730264005, −4.53121116695540434977518117393, −3.80262046352314532635788470346, −3.03650784455289186903572829244, −1.90334398580523753014845680352, −0.44156962705337997371582218644, 0.44156962705337997371582218644, 1.90334398580523753014845680352, 3.03650784455289186903572829244, 3.80262046352314532635788470346, 4.53121116695540434977518117393, 5.56715452265414334711730264005, 6.16938542931036424871859811240, 6.69937079791830339775835528432, 7.65573518970770487815204685208, 8.194349244145207073905482537800

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