Properties

Degree 2
Conductor $ 3 \cdot 11 \cdot 23^{2} $
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-s − 3-s − 4-s + 2·5-s − 6-s − 4·7-s − 3·8-s + 9-s + 2·10-s − 11-s + 12-s − 2·13-s − 4·14-s − 2·15-s − 16-s + 2·17-s + 18-s − 2·20-s + 4·21-s − 22-s + 3·24-s − 25-s − 2·26-s − 27-s + 4·28-s − 6·29-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.554·13-s − 1.06·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.447·20-s + 0.872·21-s − 0.213·22-s + 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 17457 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 17457 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(17457\)    =    \(3 \cdot 11 \cdot 23^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{17457} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 17457,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5722956163$
$L(\frac12)$  $\approx$  $0.5722956163$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;23\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;23\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
11 \( 1 + T \)
23 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−15.67670276289191, −15.41683607326312, −14.53402752733817, −14.16670416481480, −13.41166770660656, −13.17508723344350, −12.58015089406968, −12.30013932460740, −11.57721849670245, −10.68840360243007, −10.18696278925894, −9.582070429337255, −9.334439177640090, −8.684862879703934, −7.587168812136207, −7.060234579246408, −6.279075495554188, −5.786782536432515, −5.468155400475509, −4.753613168561721, −3.858616161173533, −3.359870956490712, −2.603757965429791, −1.652802520486215, −0.2856987873801149, 0.2856987873801149, 1.652802520486215, 2.603757965429791, 3.359870956490712, 3.858616161173533, 4.753613168561721, 5.468155400475509, 5.786782536432515, 6.279075495554188, 7.060234579246408, 7.587168812136207, 8.684862879703934, 9.334439177640090, 9.582070429337255, 10.18696278925894, 10.68840360243007, 11.57721849670245, 12.30013932460740, 12.58015089406968, 13.17508723344350, 13.41166770660656, 14.16670416481480, 14.53402752733817, 15.41683607326312, 15.67670276289191

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