Properties

Label 2-4560-1.1-c1-0-44
Degree $2$
Conductor $4560$
Sign $1$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s + 2·7-s + 9-s + 6·13-s + 15-s + 8·17-s − 19-s + 2·21-s + 4·23-s + 25-s + 27-s + 2·29-s + 2·31-s + 2·35-s − 2·37-s + 6·39-s − 12·41-s − 4·43-s + 45-s − 12·47-s − 3·49-s + 8·51-s + 10·53-s − 57-s − 6·59-s − 14·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.66·13-s + 0.258·15-s + 1.94·17-s − 0.229·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.359·31-s + 0.338·35-s − 0.328·37-s + 0.960·39-s − 1.87·41-s − 0.609·43-s + 0.149·45-s − 1.75·47-s − 3/7·49-s + 1.12·51-s + 1.37·53-s − 0.132·57-s − 0.781·59-s − 1.79·61-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: yes
Arithmetic: yes
Character: $\chi_{4560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.544151118\)
\(L(\frac12)\) \(\approx\) \(3.544151118\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.415428400482578942090198569994, −7.79967030739182165297416414930, −6.89573762489701331649662372428, −6.14482628670242712182783864249, −5.35202554722267828643563161324, −4.66508378245664212151193912990, −3.52246684587337315702191375794, −3.10415002597987749492791135967, −1.72066473704920085648614313077, −1.18862103578862569311714145741, 1.18862103578862569311714145741, 1.72066473704920085648614313077, 3.10415002597987749492791135967, 3.52246684587337315702191375794, 4.66508378245664212151193912990, 5.35202554722267828643563161324, 6.14482628670242712182783864249, 6.89573762489701331649662372428, 7.79967030739182165297416414930, 8.415428400482578942090198569994

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