Properties

Label 2-2280-1.1-c1-0-11
Degree $2$
Conductor $2280$
Sign $1$
Analytic cond. $18.2058$
Root an. cond. $4.26683$
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 + 9-s + 4·11-s + 2·13-s − 15-s + 6·17-s − 19-s − 4·23-s + 25-s − 27-s − 10·29-s + 4·31-s − 4·33-s + 10·37-s − 2·39-s + 6·41-s + 45-s − 12·47-s − 7·49-s − 6·51-s + 10·53-s + 4·55-s + 57-s − 10·61-s + 2·65-s + 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.229·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s − 0.696·33-s + 1.64·37-s − 0.320·39-s + 0.937·41-s + 0.149·45-s − 1.75·47-s − 49-s − 0.840·51-s + 1.37·53-s + 0.539·55-s + 0.132·57-s − 1.28·61-s + 0.248·65-s + 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2280\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(18.2058\)
Root analytic conductor: \(4.26683\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2280,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.827873442\)
\(L(\frac12)\) \(\approx\) \(1.827873442\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−9.287180121684984784965900213765, −8.147138876990079866417523759053, −7.50281712761657102889086692836, −6.35198931281804077801342630786, −6.08317621586618125737703438550, −5.16940849835265890817106579659, −4.13362534201749987180283134496, −3.39156409204262818849202841222, −1.94416852136943820176158401884, −0.962041023238845099053363878952, 0.962041023238845099053363878952, 1.94416852136943820176158401884, 3.39156409204262818849202841222, 4.13362534201749987180283134496, 5.16940849835265890817106579659, 6.08317621586618125737703438550, 6.35198931281804077801342630786, 7.50281712761657102889086692836, 8.147138876990079866417523759053, 9.287180121684984784965900213765

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