Properties

Degree 2
Conductor $ 3^{2} \cdot 17 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 5-s − 2·7-s + 2·10-s − 3·11-s − 5·13-s + 4·14-s − 4·16-s − 17-s − 19-s − 2·20-s + 6·22-s − 7·23-s − 4·25-s + 10·26-s − 4·28-s + 6·29-s + 4·31-s + 8·32-s + 2·34-s + 2·35-s + 10·37-s + 2·38-s + 9·41-s + 43-s − 6·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s − 0.755·7-s + 0.632·10-s − 0.904·11-s − 1.38·13-s + 1.06·14-s − 16-s − 0.242·17-s − 0.229·19-s − 0.447·20-s + 1.27·22-s − 1.45·23-s − 4/5·25-s + 1.96·26-s − 0.755·28-s + 1.11·29-s + 0.718·31-s + 1.41·32-s + 0.342·34-s + 0.338·35-s + 1.64·37-s + 0.324·38-s + 1.40·41-s + 0.152·43-s − 0.904·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(153\)    =    \(3^{2} \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{153} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 153,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
17 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−19.56815900582191, −19.00767291644745, −17.92217710837728, −17.42298647743133, −16.11381661835285, −15.96576130463482, −14.59963416478023, −13.32246804118300, −12.29040567291484, −11.21201107323356, −9.992759799763264, −9.673767858002872, −8.193497167062058, −7.627174360157962, −6.353218428593448, −4.540094212681588, −2.527482380710219, 0, 2.527482380710219, 4.540094212681588, 6.353218428593448, 7.627174360157962, 8.193497167062058, 9.673767858002872, 9.992759799763264, 11.21201107323356, 12.29040567291484, 13.32246804118300, 14.59963416478023, 15.96576130463482, 16.11381661835285, 17.42298647743133, 17.92217710837728, 19.00767291644745, 19.56815900582191

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