Properties

Label 2-4560-1.1-c1-0-17
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 + 4.73·7-s + 9-s − 4.19·11-s − 1.26·13-s − 15-s − 6.92·17-s − 19-s − 4.73·21-s + 6·23-s + 25-s − 27-s − 10.1·29-s + 2·31-s + 4.19·33-s + 4.73·35-s + 2.73·37-s + 1.26·39-s + 11.6·41-s + 4.73·43-s + 45-s + 10·47-s + 15.3·49-s + 6.92·51-s − 4.19·55-s + 57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.78·7-s + 0.333·9-s − 1.26·11-s − 0.351·13-s − 0.258·15-s − 1.68·17-s − 0.229·19-s − 1.03·21-s + 1.25·23-s + 0.200·25-s − 0.192·27-s − 1.89·29-s + 0.359·31-s + 0.730·33-s + 0.799·35-s + 0.449·37-s + 0.203·39-s + 1.82·41-s + 0.721·43-s + 0.149·45-s + 1.45·47-s + 2.19·49-s + 0.970·51-s − 0.565·55-s + 0.132·57-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

\(L(1)\) \(\approx\) \(1.840002862\)
\(L(\frac12)\) \(\approx\) \(1.840002862\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.73T + 7T^{2} \)
11 \( 1 + 4.19T + 11T^{2} \)
13 \( 1 + 1.26T + 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 2.73T + 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 - 4.73T + 43T^{2} \)
47 \( 1 - 10T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 9.46T + 59T^{2} \)
61 \( 1 - 5.46T + 61T^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 0.928T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + 7.66T + 89T^{2} \)
97 \( 1 - 1.66T + 97T^{2} \)
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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

−8.328761330012303401628125071910, −7.45746371869388320442434644017, −7.10261496941497293184373385082, −5.87384596442732967011460762441, −5.38297948864944214045115863347, −4.70702086601384250192774259482, −4.14630576128860509960993824945, −2.49300375073771484602837124552, −2.05706948783891339377207549974, −0.77798017491722243677583252580, 0.77798017491722243677583252580, 2.05706948783891339377207549974, 2.49300375073771484602837124552, 4.14630576128860509960993824945, 4.70702086601384250192774259482, 5.38297948864944214045115863347, 5.87384596442732967011460762441, 7.10261496941497293184373385082, 7.45746371869388320442434644017, 8.328761330012303401628125071910

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