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  − 2.41·2-s + 3.82·4-s − 5-s − 7-s − 4.41·8-s + 2.41·10-s − 0.414·11-s + 2.41·13-s + 2.41·14-s + 2.99·16-s − 4.82·17-s + 1.58·19-s − 3.82·20-s + 0.999·22-s + 0.828·23-s + 25-s − 5.82·26-s − 3.82·28-s + 4·29-s + 6·31-s + 1.58·32-s + 11.6·34-s + 35-s − 8.48·37-s − 3.82·38-s + 4.41·40-s + 7.82·41-s + ⋯
L(s)  = 1  − 1.70·2-s + 1.91·4-s − 0.447·5-s − 0.377·7-s − 1.56·8-s + 0.763·10-s − 0.124·11-s + 0.669·13-s + 0.645·14-s + 0.749·16-s − 1.17·17-s + 0.363·19-s − 0.856·20-s + 0.213·22-s + 0.172·23-s + 0.200·25-s − 1.14·26-s − 0.723·28-s + 0.742·29-s + 1.07·31-s + 0.280·32-s + 1.99·34-s + 0.169·35-s − 1.39·37-s − 0.621·38-s + 0.697·40-s + 1.22·41-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  =  $(2,\ 945,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.538988$
$L(\frac12)$  $\approx$  $0.538988$
$L(\frac{3}{2})$   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 \( 1 + 2.41T + 2T^{2} \)
11 \( 1 + 0.414T + 11T^{2} \)
13 \( 1 - 2.41T + 13T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 - 1.58T + 19T^{2} \)
23 \( 1 - 0.828T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 8.48T + 37T^{2} \)
41 \( 1 - 7.82T + 41T^{2} \)
43 \( 1 + 6.65T + 43T^{2} \)
47 \( 1 + 7.48T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 6.48T + 59T^{2} \)
61 \( 1 + 8.48T + 61T^{2} \)
67 \( 1 - 7.82T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 2.07T + 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 - 7.48T + 83T^{2} \)
89 \( 1 - 8.65T + 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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

−10.02792358613329028896778447612, −9.027239797356005786129952452974, −8.572370578576987369747785803198, −7.75991674089887446705496511930, −6.86695097279856306167237730752, −6.26244710271478449709489221025, −4.75650797871978601491027775130, −3.39046012594370553455550625182, −2.16753262769038523831216053910, −0.73557699812010142157301659130, 0.73557699812010142157301659130, 2.16753262769038523831216053910, 3.39046012594370553455550625182, 4.75650797871978601491027775130, 6.26244710271478449709489221025, 6.86695097279856306167237730752, 7.75991674089887446705496511930, 8.572370578576987369747785803198, 9.027239797356005786129952452974, 10.02792358613329028896778447612

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