Properties

Degree 2
Conductor $ 3 \cdot 5^{2} \cdot 7 \cdot 13^{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 − 6-s + 7-s − 3·8-s + 9-s + 12-s + 14-s − 16-s − 2·17-s + 18-s + 8·19-s − 21-s − 8·23-s + 3·24-s − 27-s − 28-s − 2·29-s − 4·31-s + 5·32-s − 2·34-s − 36-s − 2·37-s + 8·38-s + 6·41-s − 42-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s − 0.218·21-s − 1.66·23-s + 0.612·24-s − 0.192·27-s − 0.188·28-s − 0.371·29-s − 0.718·31-s + 0.883·32-s − 0.342·34-s − 1/6·36-s − 0.328·37-s + 1.29·38-s + 0.937·41-s − 0.154·42-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 88725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 88725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(88725\)    =    \(3 \cdot 5^{2} \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{88725} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 88725,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.454543785$
$L(\frac12)$  $\approx$  $1.454543785$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−13.88589701243103, −13.54394300464009, −12.83485558884757, −12.34300847048791, −12.11556175745082, −11.42740135963957, −11.16583699612448, −10.45201601803979, −9.859710635230275, −9.387590617728637, −9.074133711553371, −8.184618756535609, −7.866624848605683, −7.235388311199307, −6.598765383132117, −5.931022551457820, −5.610060524046339, −5.074383888265579, −4.597590563386227, −3.926518420698503, −3.570957431614462, −2.790051821589359, −2.013450548131209, −1.232333130258997, −0.3804922007573067, 0.3804922007573067, 1.232333130258997, 2.013450548131209, 2.790051821589359, 3.570957431614462, 3.926518420698503, 4.597590563386227, 5.074383888265579, 5.610060524046339, 5.931022551457820, 6.598765383132117, 7.235388311199307, 7.866624848605683, 8.184618756535609, 9.074133711553371, 9.387590617728637, 9.859710635230275, 10.45201601803979, 11.16583699612448, 11.42740135963957, 12.11556175745082, 12.34300847048791, 12.83485558884757, 13.54394300464009, 13.88589701243103

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