Properties

Label 2-4560-1.1-c1-0-13
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 − 3.83·7-s + 9-s + 1.14·11-s − 3.53·13-s + 15-s − 6.97·17-s + 19-s − 3.83·21-s + 8.97·23-s + 25-s + 27-s + 0.853·29-s + 4.39·31-s + 1.14·33-s − 3.83·35-s + 1.83·37-s − 3.53·39-s − 8.51·41-s + 10.1·43-s + 45-s − 0.978·47-s + 7.68·49-s − 6.97·51-s + 6.97·53-s + 1.14·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.44·7-s + 0.333·9-s + 0.345·11-s − 0.981·13-s + 0.258·15-s − 1.69·17-s + 0.229·19-s − 0.836·21-s + 1.87·23-s + 0.200·25-s + 0.192·27-s + 0.158·29-s + 0.789·31-s + 0.199·33-s − 0.647·35-s + 0.301·37-s − 0.566·39-s − 1.33·41-s + 1.54·43-s + 0.149·45-s − 0.142·47-s + 1.09·49-s − 0.977·51-s + 0.958·53-s + 0.154·55-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

\(L(1)\) \(\approx\) \(1.969877667\)
\(L(\frac12)\) \(\approx\) \(1.969877667\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.83T + 7T^{2} \)
11 \( 1 - 1.14T + 11T^{2} \)
13 \( 1 + 3.53T + 13T^{2} \)
17 \( 1 + 6.97T + 17T^{2} \)
23 \( 1 - 8.97T + 23T^{2} \)
29 \( 1 - 0.853T + 29T^{2} \)
31 \( 1 - 4.39T + 31T^{2} \)
37 \( 1 - 1.83T + 37T^{2} \)
41 \( 1 + 8.51T + 41T^{2} \)
43 \( 1 - 10.1T + 43T^{2} \)
47 \( 1 + 0.978T + 47T^{2} \)
53 \( 1 - 6.97T + 53T^{2} \)
59 \( 1 - 6.29T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 + 0.585T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 - 7.37T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 7.93T + 89T^{2} \)
97 \( 1 - 1.24T + 97T^{2} \)
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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

−8.623363505951479634316731172432, −7.41924121037263209694401441584, −6.77621103687427947642378131460, −6.46947821770975578865892590576, −5.33946185409008379901174344745, −4.56447920738468185414852580008, −3.64451772136082626038309899625, −2.78236509784746885664175478420, −2.25073185862266782942922809410, −0.73685881756563129953925303130, 0.73685881756563129953925303130, 2.25073185862266782942922809410, 2.78236509784746885664175478420, 3.64451772136082626038309899625, 4.56447920738468185414852580008, 5.33946185409008379901174344745, 6.46947821770975578865892590576, 6.77621103687427947642378131460, 7.41924121037263209694401441584, 8.623363505951479634316731172432

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