↑

Properties

Label 2-1680-420.419-c0-0-10
Degree $2$
Conductor $1680$
Sign $1$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 7-s + 9-s + 15-s − 21-s + 25-s + 27-s − 35-s − 2·41-s + 2·43-s + 45-s − 2·47-s + 49-s − 63-s − 2·67-s + 75-s + 81-s − 2·83-s + 2·89-s − 2·101-s − 105-s − 2·109-s + ⋯
L(s)  = 1  + 3-s + 5-s − 7-s + 9-s + 15-s − 21-s + 25-s + 27-s − 35-s − 2·41-s + 2·43-s + 45-s − 2·47-s + 49-s − 63-s − 2·67-s + 75-s + 81-s − 2·83-s + 2·89-s − 2·101-s − 105-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1680} (1679, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.713650301\)
\(L(\frac12)\) \(\approx\) \(1.713650301\)
\(L(1)\) 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 \( 1 - T \)
7 \( 1 + T \)
good11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 + T )^{2} \)
43 \( ( 1 - T )^{2} \)
47 \( ( 1 + T )^{2} \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 + T )^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 + T )^{2} \)
89 \( ( 1 - T )^{2} \)
97 \( 1 + T^{2} \)
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

−9.472046167883625039644653877451, −8.945603530536641958839738365786, −8.089712012370944163294756561211, −7.06843295039241849199355012227, −6.47562437765883722564975724771, −5.55642465345471739066920195508, −4.46362585102897919927189279131, −3.36432744990175912605880898222, −2.65779068792870325911416746662, −1.57104991769967852176906065628, 1.57104991769967852176906065628, 2.65779068792870325911416746662, 3.36432744990175912605880898222, 4.46362585102897919927189279131, 5.55642465345471739066920195508, 6.47562437765883722564975724771, 7.06843295039241849199355012227, 8.089712012370944163294756561211, 8.945603530536641958839738365786, 9.472046167883625039644653877451

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