Properties

Degree 2
Conductor $ 3^{3} \cdot 5 \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.30·2-s + 3.30·4-s + 5-s + 7-s + 3.00·8-s + 2.30·10-s + 3·11-s + 1.30·13-s + 2.30·14-s + 0.302·16-s − 2.30·17-s − 3.30·19-s + 3.30·20-s + 6.90·22-s + 0.697·23-s + 25-s + 3·26-s + 3.30·28-s + 5.30·29-s − 2.39·31-s − 5.30·32-s − 5.30·34-s + 35-s + 3.60·37-s − 7.60·38-s + 3.00·40-s − 6.90·41-s + ⋯
L(s)  = 1  + 1.62·2-s + 1.65·4-s + 0.447·5-s + 0.377·7-s + 1.06·8-s + 0.728·10-s + 0.904·11-s + 0.361·13-s + 0.615·14-s + 0.0756·16-s − 0.558·17-s − 0.757·19-s + 0.738·20-s + 1.47·22-s + 0.145·23-s + 0.200·25-s + 0.588·26-s + 0.624·28-s + 0.984·29-s − 0.430·31-s − 0.937·32-s − 0.909·34-s + 0.169·35-s + 0.592·37-s − 1.23·38-s + 0.474·40-s − 1.07·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{945} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 945,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.39936$
$L(\frac12)$  $\approx$  $4.39936$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
good2 \( 1 - 2.30T + 2T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - 1.30T + 13T^{2} \)
17 \( 1 + 2.30T + 17T^{2} \)
19 \( 1 + 3.30T + 19T^{2} \)
23 \( 1 - 0.697T + 23T^{2} \)
29 \( 1 - 5.30T + 29T^{2} \)
31 \( 1 + 2.39T + 31T^{2} \)
37 \( 1 - 3.60T + 37T^{2} \)
41 \( 1 + 6.90T + 41T^{2} \)
43 \( 1 + 4.21T + 43T^{2} \)
47 \( 1 - 7.60T + 47T^{2} \)
53 \( 1 + 5.51T + 53T^{2} \)
59 \( 1 - 6.21T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 13.5T + 67T^{2} \)
71 \( 1 - 2.09T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 + 12.2T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + 9.51T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.32311307127426569087854726205, −9.168744348213449786245213795433, −8.379419098498571764255089098175, −7.00208684727925169722721474597, −6.41516095465803900924521091338, −5.61599385168131930874559514651, −4.64584271661518399921731908072, −3.97193850209979659721996294033, −2.83922855426164383936617844541, −1.69463971121385263586712241401, 1.69463971121385263586712241401, 2.83922855426164383936617844541, 3.97193850209979659721996294033, 4.64584271661518399921731908072, 5.61599385168131930874559514651, 6.41516095465803900924521091338, 7.00208684727925169722721474597, 8.379419098498571764255089098175, 9.168744348213449786245213795433, 10.32311307127426569087854726205

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