Properties

Label 2-4560-1.1-c1-0-52
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 0.732·7-s + 9-s − 1.26·11-s − 2.73·13-s − 15-s − 19-s + 0.732·21-s + 3.46·23-s + 25-s − 27-s − 2.19·29-s + 4.92·31-s + 1.26·33-s − 0.732·35-s + 4.19·37-s + 2.73·39-s + 4.73·41-s + 6.19·43-s + 45-s − 3.46·47-s − 6.46·49-s − 2.53·53-s − 1.26·55-s + 57-s − 9.46·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.276·7-s + 0.333·9-s − 0.382·11-s − 0.757·13-s − 0.258·15-s − 0.229·19-s + 0.159·21-s + 0.722·23-s + 0.200·25-s − 0.192·27-s − 0.407·29-s + 0.885·31-s + 0.220·33-s − 0.123·35-s + 0.689·37-s + 0.437·39-s + 0.739·41-s + 0.944·43-s + 0.149·45-s − 0.505·47-s − 0.923·49-s − 0.348·53-s − 0.170·55-s + 0.132·57-s − 1.23·59-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: \(1\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.732T + 7T^{2} \)
11 \( 1 + 1.26T + 11T^{2} \)
13 \( 1 + 2.73T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 2.19T + 29T^{2} \)
31 \( 1 - 4.92T + 31T^{2} \)
37 \( 1 - 4.19T + 37T^{2} \)
41 \( 1 - 4.73T + 41T^{2} \)
43 \( 1 - 6.19T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 + 2.53T + 53T^{2} \)
59 \( 1 + 9.46T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 16.3T + 71T^{2} \)
73 \( 1 + 3.07T + 73T^{2} \)
79 \( 1 + 2.92T + 79T^{2} \)
83 \( 1 + 0.928T + 83T^{2} \)
89 \( 1 - 7.26T + 89T^{2} \)
97 \( 1 - 4.19T + 97T^{2} \)
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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

−7.71100449868886700597569809564, −7.33881018523260123913871389899, −6.24908694924913266206824850002, −5.98373620210564573161353804353, −4.89209556028868913419168897654, −4.51342957722350334999320583339, −3.22188625947209696764156220730, −2.45703976040872182094071098321, −1.31177973225248355716976267477, 0, 1.31177973225248355716976267477, 2.45703976040872182094071098321, 3.22188625947209696764156220730, 4.51342957722350334999320583339, 4.89209556028868913419168897654, 5.98373620210564573161353804353, 6.24908694924913266206824850002, 7.33881018523260123913871389899, 7.71100449868886700597569809564

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