Properties

Label 2-2280-1.1-c1-0-26
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s − 2.52·7-s + 9-s − 3.10·11-s + 6.72·13-s − 15-s − 2.57·17-s − 19-s + 2.52·21-s − 4.57·23-s + 25-s − 27-s − 1.10·29-s + 7.83·31-s + 3.10·33-s − 2.52·35-s − 4.52·37-s − 6.72·39-s + 6.15·41-s − 7.68·43-s + 45-s − 3.42·47-s − 0.627·49-s + 2.57·51-s + 2.57·53-s − 3.10·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.954·7-s + 0.333·9-s − 0.935·11-s + 1.86·13-s − 0.258·15-s − 0.625·17-s − 0.229·19-s + 0.550·21-s − 0.954·23-s + 0.200·25-s − 0.192·27-s − 0.204·29-s + 1.40·31-s + 0.540·33-s − 0.426·35-s − 0.743·37-s − 1.07·39-s + 0.960·41-s − 1.17·43-s + 0.149·45-s − 0.499·47-s − 0.0896·49-s + 0.361·51-s + 0.354·53-s − 0.418·55-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: 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 \( 1 + 2.52T + 7T^{2} \)
11 \( 1 + 3.10T + 11T^{2} \)
13 \( 1 - 6.72T + 13T^{2} \)
17 \( 1 + 2.57T + 17T^{2} \)
23 \( 1 + 4.57T + 23T^{2} \)
29 \( 1 + 1.10T + 29T^{2} \)
31 \( 1 - 7.83T + 31T^{2} \)
37 \( 1 + 4.52T + 37T^{2} \)
41 \( 1 - 6.15T + 41T^{2} \)
43 \( 1 + 7.68T + 43T^{2} \)
47 \( 1 + 3.42T + 47T^{2} \)
53 \( 1 - 2.57T + 53T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 - 8.41T + 67T^{2} \)
71 \( 1 - 9.45T + 71T^{2} \)
73 \( 1 + 9.25T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4.47T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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

−8.610750011138441833014484071164, −7.963354191479341623732652322253, −6.77230316278390298223509126641, −6.23854812555862474028484122781, −5.71395450387227694597263214000, −4.65654298619858796014788257304, −3.70376367865501333865061227317, −2.75046929565853231880203129369, −1.48291588483107598247836553906, 0, 1.48291588483107598247836553906, 2.75046929565853231880203129369, 3.70376367865501333865061227317, 4.65654298619858796014788257304, 5.71395450387227694597263214000, 6.23854812555862474028484122781, 6.77230316278390298223509126641, 7.963354191479341623732652322253, 8.610750011138441833014484071164

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