Properties

Label 2-2280-1.1-c1-0-34
Degree $2$
Conductor $2280$
Sign $-1$
Analytic cond. $18.2058$
Root an. cond. $4.26683$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 1.26·7-s + 9-s − 6.19·11-s − 4.73·13-s + 15-s + 6.92·17-s + 19-s − 1.26·21-s − 6·23-s + 25-s + 27-s + 0.196·29-s − 2·31-s − 6.19·33-s − 1.26·35-s − 0.732·37-s − 4.73·39-s − 5.66·41-s − 1.26·43-s + 45-s − 10·47-s − 5.39·49-s + 6.92·51-s − 6.19·55-s + 57-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.479·7-s + 0.333·9-s − 1.86·11-s − 1.31·13-s + 0.258·15-s + 1.68·17-s + 0.229·19-s − 0.276·21-s − 1.25·23-s + 0.200·25-s + 0.192·27-s + 0.0364·29-s − 0.359·31-s − 1.07·33-s − 0.214·35-s − 0.120·37-s − 0.757·39-s − 0.883·41-s − 0.193·43-s + 0.149·45-s − 1.45·47-s − 0.770·49-s + 0.970·51-s − 0.835·55-s + 0.132·57-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: no
Arithmetic: yes
Character: $\chi_{2280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2280,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.26T + 7T^{2} \)
11 \( 1 + 6.19T + 11T^{2} \)
13 \( 1 + 4.73T + 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 - 0.196T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 0.732T + 37T^{2} \)
41 \( 1 + 5.66T + 41T^{2} \)
43 \( 1 + 1.26T + 43T^{2} \)
47 \( 1 + 10T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 2.53T + 59T^{2} \)
61 \( 1 + 1.46T + 61T^{2} \)
67 \( 1 + 1.07T + 67T^{2} \)
71 \( 1 + 8.39T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 15.4T + 83T^{2} \)
89 \( 1 - 9.66T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−8.538688904736345477336043619962, −7.71948740898332032461388229645, −7.44128621014626699801732547193, −6.22477819396715579764507624959, −5.37551825403167024275416374702, −4.78862304978470796685144678068, −3.38718833782023405178767971351, −2.78645411329409699112232096660, −1.82377143662330237664447438244, 0, 1.82377143662330237664447438244, 2.78645411329409699112232096660, 3.38718833782023405178767971351, 4.78862304978470796685144678068, 5.37551825403167024275416374702, 6.22477819396715579764507624959, 7.44128621014626699801732547193, 7.71948740898332032461388229645, 8.538688904736345477336043619962

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