Properties

Label 2-4560-1.1-c1-0-51
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 + 1.41·7-s + 9-s + 2.24·11-s − 3.41·13-s + 15-s + 1.17·17-s + 19-s − 1.41·21-s − 7.65·23-s + 25-s − 27-s + 1.41·29-s + 3.17·31-s − 2.24·33-s − 1.41·35-s − 3.41·37-s + 3.41·39-s − 0.242·41-s − 12.2·43-s − 45-s + 7.65·47-s − 5·49-s − 1.17·51-s + 8·53-s − 2.24·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.534·7-s + 0.333·9-s + 0.676·11-s − 0.946·13-s + 0.258·15-s + 0.284·17-s + 0.229·19-s − 0.308·21-s − 1.59·23-s + 0.200·25-s − 0.192·27-s + 0.262·29-s + 0.569·31-s − 0.390·33-s − 0.239·35-s − 0.561·37-s + 0.546·39-s − 0.0378·41-s − 1.86·43-s − 0.149·45-s + 1.11·47-s − 0.714·49-s − 0.164·51-s + 1.09·53-s − 0.302·55-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: \(1\)
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 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 - 2.24T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
23 \( 1 + 7.65T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 3.17T + 31T^{2} \)
37 \( 1 + 3.41T + 37T^{2} \)
41 \( 1 + 0.242T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 - 7.65T + 47T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 7.31T + 61T^{2} \)
67 \( 1 - 9.65T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 7.65T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 - 5.89T + 89T^{2} \)
97 \( 1 + 9.75T + 97T^{2} \)
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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

−8.000044225901767224227961642725, −7.17848085074397315353680118348, −6.59049727721469305012632514707, −5.70646453267605422761103764100, −4.97975539405571724413945817985, −4.28789627647524852567730305147, −3.51832248531185203347291894514, −2.31678414121467265869669339776, −1.30286531081669497257238006184, 0, 1.30286531081669497257238006184, 2.31678414121467265869669339776, 3.51832248531185203347291894514, 4.28789627647524852567730305147, 4.97975539405571724413945817985, 5.70646453267605422761103764100, 6.59049727721469305012632514707, 7.17848085074397315353680118348, 8.000044225901767224227961642725

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