Properties

Label 2-2280-1.1-c1-0-35
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 2·7-s + 9-s − 2·11-s − 4·13-s + 15-s − 2·17-s − 19-s − 2·21-s + 25-s + 27-s − 4·29-s + 8·31-s − 2·33-s − 2·35-s − 8·37-s − 4·39-s − 12·41-s − 6·43-s + 45-s − 3·49-s − 2·51-s − 2·53-s − 2·55-s − 57-s + 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.258·15-s − 0.485·17-s − 0.229·19-s − 0.436·21-s + 1/5·25-s + 0.192·27-s − 0.742·29-s + 1.43·31-s − 0.348·33-s − 0.338·35-s − 1.31·37-s − 0.640·39-s − 1.87·41-s − 0.914·43-s + 0.149·45-s − 3/7·49-s − 0.280·51-s − 0.274·53-s − 0.269·55-s − 0.132·57-s + 0.256·61-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.635560404135754740170429774671, −7.960636380029441787969451220790, −6.97964208027671263402172640499, −6.51093236128179904507084466386, −5.36056849345200491493259310620, −4.68224220184262448853244509134, −3.49785400828546454141149719639, −2.71571760667042952942857818313, −1.83542406864400221341156505183, 0, 1.83542406864400221341156505183, 2.71571760667042952942857818313, 3.49785400828546454141149719639, 4.68224220184262448853244509134, 5.36056849345200491493259310620, 6.51093236128179904507084466386, 6.97964208027671263402172640499, 7.960636380029441787969451220790, 8.635560404135754740170429774671

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