Properties

Label 2-4560-1.1-c1-0-45
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 + 3.64·7-s + 9-s + 5.64·11-s − 1.64·13-s + 15-s + 19-s + 3.64·21-s − 2·23-s + 25-s + 27-s + 4.35·29-s + 2·31-s + 5.64·33-s + 3.64·35-s + 8.93·37-s − 1.64·39-s − 3.64·41-s − 10.9·43-s + 45-s − 6·47-s + 6.29·49-s + 4·53-s + 5.64·55-s + 57-s + 7.29·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.37·7-s + 0.333·9-s + 1.70·11-s − 0.456·13-s + 0.258·15-s + 0.229·19-s + 0.795·21-s − 0.417·23-s + 0.200·25-s + 0.192·27-s + 0.808·29-s + 0.359·31-s + 0.982·33-s + 0.616·35-s + 1.46·37-s − 0.263·39-s − 0.569·41-s − 1.66·43-s + 0.149·45-s − 0.875·47-s + 0.898·49-s + 0.549·53-s + 0.761·55-s + 0.132·57-s + 0.949·59-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: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.658989202\)
\(L(\frac12)\) \(\approx\) \(3.658989202\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.64T + 7T^{2} \)
11 \( 1 - 5.64T + 11T^{2} \)
13 \( 1 + 1.64T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 - 4.35T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 8.93T + 37T^{2} \)
41 \( 1 + 3.64T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 7.29T + 59T^{2} \)
61 \( 1 - 0.708T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 7.29T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 8.22T + 97T^{2} \)
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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

−8.378309426815539897520853714466, −7.74741352423048493119381581866, −6.86459974175772536417860130436, −6.28668354697129778483231024563, −5.24226187610104879422397118322, −4.55045502353599921608921653401, −3.88111592719966943130413809287, −2.80265042955694524879887531705, −1.81118068223647028460996115851, −1.18900520423033290457766561742, 1.18900520423033290457766561742, 1.81118068223647028460996115851, 2.80265042955694524879887531705, 3.88111592719966943130413809287, 4.55045502353599921608921653401, 5.24226187610104879422397118322, 6.28668354697129778483231024563, 6.86459974175772536417860130436, 7.74741352423048493119381581866, 8.378309426815539897520853714466

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