Properties

Degree 2
Conductor $ 2^{4} \cdot 7 \cdot 19 \cdot 47 $
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  − 3-s + 5-s + 7-s − 2·9-s + 2·11-s + 4·13-s − 15-s − 6·17-s − 19-s − 21-s − 6·23-s − 4·25-s + 5·27-s − 7·29-s − 2·31-s − 2·33-s + 35-s + 2·37-s − 4·39-s + 6·41-s + 6·43-s − 2·45-s + 47-s + 49-s + 6·51-s + 2·55-s + 57-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.603·11-s + 1.10·13-s − 0.258·15-s − 1.45·17-s − 0.229·19-s − 0.218·21-s − 1.25·23-s − 4/5·25-s + 0.962·27-s − 1.29·29-s − 0.359·31-s − 0.348·33-s + 0.169·35-s + 0.328·37-s − 0.640·39-s + 0.937·41-s + 0.914·43-s − 0.298·45-s + 0.145·47-s + 1/7·49-s + 0.840·51-s + 0.269·55-s + 0.132·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100016\)    =    \(2^{4} \cdot 7 \cdot 19 \cdot 47\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100016} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 100016,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;19,\;47\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;19,\;47\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 - T \)
19 \( 1 + T \)
47 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−13.99369218027643, −13.55059365284728, −13.04868776834304, −12.55353392037402, −11.89490194764440, −11.41705770707108, −11.19607956020717, −10.68082236318332, −10.21128212311022, −9.406925975791239, −8.987575565428963, −8.749619259108616, −7.914533708089427, −7.597924183309416, −6.736139074976686, −6.238126809926264, −5.875087509843079, −5.591135294425248, −4.655038252396157, −4.213528098367691, −3.716994095197134, −2.916062397852739, −2.031604518865628, −1.805247746884937, −0.7940724262743462, 0, 0.7940724262743462, 1.805247746884937, 2.031604518865628, 2.916062397852739, 3.716994095197134, 4.213528098367691, 4.655038252396157, 5.591135294425248, 5.875087509843079, 6.238126809926264, 6.736139074976686, 7.597924183309416, 7.914533708089427, 8.749619259108616, 8.987575565428963, 9.406925975791239, 10.21128212311022, 10.68082236318332, 11.19607956020717, 11.41705770707108, 11.89490194764440, 12.55353392037402, 13.04868776834304, 13.55059365284728, 13.99369218027643

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