Properties

Degree 2
Conductor $ 3 \cdot 11 \cdot 43^{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 + 8·23-s + 3·24-s − 25-s + 2·26-s + 27-s + 4·28-s + 6·29-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 + 1.66·23-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 + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 61017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 61017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(61017\)    =    \(3 \cdot 11 \cdot 43^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{61017} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 61017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.255657989$
$L(\frac12)$  $\approx$  $1.255657989$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;11,\;43\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;11,\;43\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
11 \( 1 - T \)
43 \( 1 \)
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 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
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

−14.12038862640822, −13.68989542065110, −13.40739473296481, −12.85692911985383, −12.50044602920096, −11.89533991933459, −10.85385417683054, −10.55267045854690, −10.03821876915576, −9.507042081803527, −9.178475186377453, −8.927889443706230, −8.312111250154402, −7.410848086026639, −7.124485114766799, −6.547973638036471, −5.949065770490848, −5.193362678088571, −4.745867806147640, −3.858007634528341, −3.412305863553091, −2.668257918888328, −2.071053446875675, −1.257709646867274, −0.4393321260479175, 0.4393321260479175, 1.257709646867274, 2.071053446875675, 2.668257918888328, 3.412305863553091, 3.858007634528341, 4.745867806147640, 5.193362678088571, 5.949065770490848, 6.547973638036471, 7.124485114766799, 7.410848086026639, 8.312111250154402, 8.927889443706230, 9.178475186377453, 9.507042081803527, 10.03821876915576, 10.55267045854690, 10.85385417683054, 11.89533991933459, 12.50044602920096, 12.85692911985383, 13.40739473296481, 13.68989542065110, 14.12038862640822

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