Properties

Label 2-4560-1.1-c1-0-11
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 + 4.21·7-s + 9-s − 5.57·11-s − 2.21·13-s + 15-s + 0.643·17-s + 19-s − 4.21·21-s − 1.35·23-s + 25-s − 27-s + 4.86·29-s − 2.64·31-s + 5.57·33-s − 4.21·35-s − 2.21·37-s + 2.21·39-s + 3.57·41-s + 0.218·43-s − 45-s + 1.35·47-s + 10.7·49-s − 0.643·51-s + 0.643·53-s + 5.57·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.59·7-s + 0.333·9-s − 1.68·11-s − 0.615·13-s + 0.258·15-s + 0.156·17-s + 0.229·19-s − 0.920·21-s − 0.282·23-s + 0.200·25-s − 0.192·27-s + 0.902·29-s − 0.474·31-s + 0.970·33-s − 0.713·35-s − 0.364·37-s + 0.355·39-s + 0.558·41-s + 0.0332·43-s − 0.149·45-s + 0.197·47-s + 1.54·49-s − 0.0900·51-s + 0.0883·53-s + 0.751·55-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

\(L(1)\) \(\approx\) \(1.353088184\)
\(L(\frac12)\) \(\approx\) \(1.353088184\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.21T + 7T^{2} \)
11 \( 1 + 5.57T + 11T^{2} \)
13 \( 1 + 2.21T + 13T^{2} \)
17 \( 1 - 0.643T + 17T^{2} \)
23 \( 1 + 1.35T + 23T^{2} \)
29 \( 1 - 4.86T + 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 + 2.21T + 37T^{2} \)
41 \( 1 - 3.57T + 41T^{2} \)
43 \( 1 - 0.218T + 43T^{2} \)
47 \( 1 - 1.35T + 47T^{2} \)
53 \( 1 - 0.643T + 53T^{2} \)
59 \( 1 - 4.43T + 59T^{2} \)
61 \( 1 - 13.0T + 61T^{2} \)
67 \( 1 + 1.28T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 - 1.28T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 - 5.78T + 97T^{2} \)
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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

−8.174665113911723343324402009095, −7.60829909035508706332576000707, −7.14895817382775261480952973757, −5.95015214540643568874222890407, −5.12633724426551357687722013577, −4.90857428746292522172906084883, −3.99384385244426311562481846001, −2.75854425724953520010649723259, −1.91836667308977616396531829929, −0.66301121612278665301158155634, 0.66301121612278665301158155634, 1.91836667308977616396531829929, 2.75854425724953520010649723259, 3.99384385244426311562481846001, 4.90857428746292522172906084883, 5.12633724426551357687722013577, 5.95015214540643568874222890407, 7.14895817382775261480952973757, 7.60829909035508706332576000707, 8.174665113911723343324402009095

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