Properties

Degree 2
Conductor $ 3^{2} \cdot 17 $
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 − 4-s + 2·5-s + 4·7-s − 3·8-s + 2·10-s − 2·13-s + 4·14-s − 16-s − 17-s − 4·19-s − 2·20-s − 4·23-s − 25-s − 2·26-s − 4·28-s − 6·29-s + 4·31-s + 5·32-s − 34-s + 8·35-s − 2·37-s − 4·38-s − 6·40-s + 6·41-s + 4·43-s − 4·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.51·7-s − 1.06·8-s + 0.632·10-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.447·20-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.755·28-s − 1.11·29-s + 0.718·31-s + 0.883·32-s − 0.171·34-s + 1.35·35-s − 0.328·37-s − 0.648·38-s − 0.948·40-s + 0.937·41-s + 0.609·43-s − 0.589·46-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  =  0
Selberg data  =  $(2,\ 153,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.585253218$
$L(\frac12)$  $\approx$  $1.585253218$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−19.29669625874846, −18.17015776245415, −17.64548380453841, −17.00770173329555, −15.40867410938629, −14.52687713459305, −14.05569027540334, −13.13887678046194, −12.16602218247269, −11.11422065032988, −9.912917364966796, −8.872689562844899, −7.818865644736769, −6.123809914785217, −5.152385804471193, −4.189768988550687, −2.145039149869537, 2.145039149869537, 4.189768988550687, 5.152385804471193, 6.123809914785217, 7.818865644736769, 8.872689562844899, 9.912917364966796, 11.11422065032988, 12.16602218247269, 13.13887678046194, 14.05569027540334, 14.52687713459305, 15.40867410938629, 17.00770173329555, 17.64548380453841, 18.17015776245415, 19.29669625874846

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