Properties

Label 2-4560-1.1-c1-0-19
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 + 1.32·7-s + 9-s − 3.70·11-s + 3.32·13-s − 15-s + 1.61·17-s + 19-s − 1.32·21-s + 7.67·23-s + 25-s − 27-s − 1.70·29-s − 9.67·31-s + 3.70·33-s + 1.32·35-s + 5.96·37-s − 3.32·39-s + 2.29·41-s + 8.73·43-s + 45-s − 11.6·47-s − 5.25·49-s − 1.61·51-s + 10.4·53-s − 3.70·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.499·7-s + 0.333·9-s − 1.11·11-s + 0.921·13-s − 0.258·15-s + 0.391·17-s + 0.229·19-s − 0.288·21-s + 1.59·23-s + 0.200·25-s − 0.192·27-s − 0.317·29-s − 1.73·31-s + 0.645·33-s + 0.223·35-s + 0.980·37-s − 0.531·39-s + 0.358·41-s + 1.33·43-s + 0.149·45-s − 1.70·47-s − 0.750·49-s − 0.225·51-s + 1.43·53-s − 0.499·55-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

\(L(1)\) \(\approx\) \(1.838590180\)
\(L(\frac12)\) \(\approx\) \(1.838590180\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.32T + 7T^{2} \)
11 \( 1 + 3.70T + 11T^{2} \)
13 \( 1 - 3.32T + 13T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
23 \( 1 - 7.67T + 23T^{2} \)
29 \( 1 + 1.70T + 29T^{2} \)
31 \( 1 + 9.67T + 31T^{2} \)
37 \( 1 - 5.96T + 37T^{2} \)
41 \( 1 - 2.29T + 41T^{2} \)
43 \( 1 - 8.73T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 - 10.4T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 - 9.02T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + 5.02T + 83T^{2} \)
89 \( 1 + 1.70T + 89T^{2} \)
97 \( 1 + 0.679T + 97T^{2} \)
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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

−8.237461006035578476351754527824, −7.59110956580293799470602896136, −6.88086063007551704425284640357, −5.98139400138620937695997937572, −5.37755533615438930855914775757, −4.87783835188731916326023182473, −3.80233888997935225496187360522, −2.87160566060930521026877425123, −1.81425456961360794284694198743, −0.804103216101608112698240117450, 0.804103216101608112698240117450, 1.81425456961360794284694198743, 2.87160566060930521026877425123, 3.80233888997935225496187360522, 4.87783835188731916326023182473, 5.37755533615438930855914775757, 5.98139400138620937695997937572, 6.88086063007551704425284640357, 7.59110956580293799470602896136, 8.237461006035578476351754527824

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