Properties

Label 2-4560-1.1-c1-0-4
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 − 4.38·7-s + 9-s − 6.38·11-s − 1.13·13-s − 15-s + 1.72·17-s − 19-s − 4.38·21-s − 1.52·23-s + 25-s + 27-s + 6.65·29-s + 3.72·31-s − 6.38·33-s + 4.38·35-s − 4.59·37-s − 1.13·39-s − 9.64·41-s − 2.59·43-s − 45-s + 9.52·47-s + 12.2·49-s + 1.72·51-s + 11.5·53-s + 6.38·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s − 1.65·7-s + 0.333·9-s − 1.92·11-s − 0.314·13-s − 0.258·15-s + 0.419·17-s − 0.229·19-s − 0.957·21-s − 0.317·23-s + 0.200·25-s + 0.192·27-s + 1.23·29-s + 0.669·31-s − 1.11·33-s + 0.741·35-s − 0.755·37-s − 0.181·39-s − 1.50·41-s − 0.395·43-s − 0.149·45-s + 1.38·47-s + 1.75·49-s + 0.242·51-s + 1.58·53-s + 0.861·55-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

\(L(1)\) \(\approx\) \(1.005637441\)
\(L(\frac12)\) \(\approx\) \(1.005637441\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.38T + 7T^{2} \)
11 \( 1 + 6.38T + 11T^{2} \)
13 \( 1 + 1.13T + 13T^{2} \)
17 \( 1 - 1.72T + 17T^{2} \)
23 \( 1 + 1.52T + 23T^{2} \)
29 \( 1 - 6.65T + 29T^{2} \)
31 \( 1 - 3.72T + 31T^{2} \)
37 \( 1 + 4.59T + 37T^{2} \)
41 \( 1 + 9.64T + 41T^{2} \)
43 \( 1 + 2.59T + 43T^{2} \)
47 \( 1 - 9.52T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 14.9T + 59T^{2} \)
61 \( 1 + 1.79T + 61T^{2} \)
67 \( 1 - 7.04T + 67T^{2} \)
71 \( 1 - 9.31T + 71T^{2} \)
73 \( 1 + 8.50T + 73T^{2} \)
79 \( 1 - 9.25T + 79T^{2} \)
83 \( 1 + 9.04T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + 4.59T + 97T^{2} \)
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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

−8.330756150970532981866916109783, −7.61679748027876702263386187870, −6.99859155621159722215605466215, −6.23820889168846001023862730722, −5.36487259819267208050322338016, −4.56101495918939957952013174343, −3.49469238893022622688385567644, −2.98219200991965002596393503180, −2.27699329572006245533550470561, −0.50618660957208422987682212753, 0.50618660957208422987682212753, 2.27699329572006245533550470561, 2.98219200991965002596393503180, 3.49469238893022622688385567644, 4.56101495918939957952013174343, 5.36487259819267208050322338016, 6.23820889168846001023862730722, 6.99859155621159722215605466215, 7.61679748027876702263386187870, 8.330756150970532981866916109783

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