Properties

Label 2-2280-1.1-c1-0-28
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s − 2·13-s − 15-s − 6·17-s + 19-s + 4·23-s + 25-s − 27-s − 6·29-s − 2·37-s + 2·39-s − 2·41-s − 4·43-s + 45-s − 4·47-s − 7·49-s + 6·51-s − 6·53-s − 57-s + 12·59-s − 2·61-s − 2·65-s + 4·67-s − 4·69-s − 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-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 − 1.11·29-s − 0.328·37-s + 0.320·39-s − 0.312·41-s − 0.609·43-s + 0.149·45-s − 0.583·47-s − 49-s + 0.840·51-s − 0.824·53-s − 0.132·57-s + 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.488·67-s − 0.481·69-s − 0.702·73-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: Trivial
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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
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

−8.768626210118887679291854915430, −7.78052147180357180110439859452, −6.87606868618011397600478902138, −6.41809015068399926337628047211, −5.33754001761415446732491834460, −4.84409972963554850352386409259, −3.78706787853046917517176527305, −2.59414462363986928995777015747, −1.57225358008099559303589986979, 0, 1.57225358008099559303589986979, 2.59414462363986928995777015747, 3.78706787853046917517176527305, 4.84409972963554850352386409259, 5.33754001761415446732491834460, 6.41809015068399926337628047211, 6.87606868618011397600478902138, 7.78052147180357180110439859452, 8.768626210118887679291854915430

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