Properties

Label 2-4560-1.1-c1-0-40
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 + 2.60·7-s + 9-s + 4.60·11-s + 4.60·13-s − 15-s + 2·17-s + 19-s + 2.60·21-s + 2·23-s + 25-s + 27-s + 2.60·29-s − 4·31-s + 4.60·33-s − 2.60·35-s + 3.39·37-s + 4.60·39-s + 6.60·41-s − 10.6·43-s − 45-s − 6·47-s − 0.211·49-s + 2·51-s − 4.60·55-s + 57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.984·7-s + 0.333·9-s + 1.38·11-s + 1.27·13-s − 0.258·15-s + 0.485·17-s + 0.229·19-s + 0.568·21-s + 0.417·23-s + 0.200·25-s + 0.192·27-s + 0.483·29-s − 0.718·31-s + 0.801·33-s − 0.440·35-s + 0.558·37-s + 0.737·39-s + 1.03·41-s − 1.61·43-s − 0.149·45-s − 0.875·47-s − 0.0301·49-s + 0.280·51-s − 0.621·55-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: 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

\(L(1)\) \(\approx\) \(3.184803204\)
\(L(\frac12)\) \(\approx\) \(3.184803204\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.60T + 7T^{2} \)
11 \( 1 - 4.60T + 11T^{2} \)
13 \( 1 - 4.60T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 - 2.60T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 - 6.60T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 5.21T + 59T^{2} \)
61 \( 1 + 7.21T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 9.21T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 - 6.60T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−8.129990907965740530384268872385, −7.939042342843346260053904486206, −6.87653934196805019528345719844, −6.30714011047980416701707001899, −5.28685499755853746652370235034, −4.43075586664445996428343849319, −3.76659040958673011292069100696, −3.08656680534505380166577694491, −1.71791492910901594510364452790, −1.10860880089444067618322377423, 1.10860880089444067618322377423, 1.71791492910901594510364452790, 3.08656680534505380166577694491, 3.76659040958673011292069100696, 4.43075586664445996428343849319, 5.28685499755853746652370235034, 6.30714011047980416701707001899, 6.87653934196805019528345719844, 7.939042342843346260053904486206, 8.129990907965740530384268872385

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