Properties

Label 2-4560-1.1-c1-0-43
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 + 4.81·7-s + 9-s + 1.45·11-s − 2.81·13-s + 15-s + 6.26·17-s − 19-s + 4.81·21-s − 6.26·23-s + 25-s + 27-s + 0.548·29-s − 8.26·31-s + 1.45·33-s + 4.81·35-s + 6.81·37-s − 2.81·39-s + 4.54·41-s + 7.71·43-s + 45-s + 10.2·47-s + 16.1·49-s + 6.26·51-s − 0.265·53-s + 1.45·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.81·7-s + 0.333·9-s + 0.437·11-s − 0.780·13-s + 0.258·15-s + 1.51·17-s − 0.229·19-s + 1.05·21-s − 1.30·23-s + 0.200·25-s + 0.192·27-s + 0.101·29-s − 1.48·31-s + 0.252·33-s + 0.813·35-s + 1.12·37-s − 0.450·39-s + 0.710·41-s + 1.17·43-s + 0.149·45-s + 1.49·47-s + 2.30·49-s + 0.877·51-s − 0.0364·53-s + 0.195·55-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

\(L(1)\) \(\approx\) \(3.560258038\)
\(L(\frac12)\) \(\approx\) \(3.560258038\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.81T + 7T^{2} \)
11 \( 1 - 1.45T + 11T^{2} \)
13 \( 1 + 2.81T + 13T^{2} \)
17 \( 1 - 6.26T + 17T^{2} \)
23 \( 1 + 6.26T + 23T^{2} \)
29 \( 1 - 0.548T + 29T^{2} \)
31 \( 1 + 8.26T + 31T^{2} \)
37 \( 1 - 6.81T + 37T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 - 7.71T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + 0.265T + 53T^{2} \)
59 \( 1 + 2.90T + 59T^{2} \)
61 \( 1 - 11.3T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 6.90T + 71T^{2} \)
73 \( 1 + 6.53T + 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 + 9.16T + 83T^{2} \)
89 \( 1 + 18.7T + 89T^{2} \)
97 \( 1 + 6.81T + 97T^{2} \)
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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

−8.296799481418709519383355564871, −7.54088359570004064873408694631, −7.31947812478263028776705568286, −5.85862737715754439959853550365, −5.50078450827291010944317078384, −4.46436373347290706958299627214, −3.96572390799856246632232345208, −2.68241167661707197994177492881, −1.93659036864651826985359685734, −1.13126660772651465550191689324, 1.13126660772651465550191689324, 1.93659036864651826985359685734, 2.68241167661707197994177492881, 3.96572390799856246632232345208, 4.46436373347290706958299627214, 5.50078450827291010944317078384, 5.85862737715754439959853550365, 7.31947812478263028776705568286, 7.54088359570004064873408694631, 8.296799481418709519383355564871

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