Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 37 \cdot 53 $
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-s + 3-s − 4-s − 2·5-s + 6-s + 2·7-s − 3·8-s + 9-s − 2·10-s − 6·11-s − 12-s + 2·13-s + 2·14-s − 2·15-s − 16-s − 17-s + 18-s − 8·19-s + 2·20-s + 2·21-s − 6·22-s + 6·23-s − 3·24-s − 25-s + 2·26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.80·11-s − 0.288·12-s + 0.554·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.83·19-s + 0.447·20-s + 0.436·21-s − 1.27·22-s + 1.25·23-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(100011\)    =    \(3 \cdot 17 \cdot 37 \cdot 53\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100011} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 100011,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;17,\;37,\;53\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;37,\;53\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
17 \( 1 + T \)
37 \( 1 + T \)
53 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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

−13.93802108703619, −13.45596189468759, −13.01015179100566, −12.67278234580962, −12.40504438064274, −11.45180113563216, −11.08493872019481, −10.72974170717674, −10.19737482589787, −9.378046536614405, −8.832141531482335, −8.539671047775760, −8.068876595491768, −7.494788827452983, −7.201777033125233, −6.202442258826556, −5.762856298770622, −5.057787947484691, −4.696233350630486, −4.210936959712628, −3.580972730771657, −3.197309307072032, −2.339331061797025, −1.949115435795926, −0.7068381036555976, 0, 0.7068381036555976, 1.949115435795926, 2.339331061797025, 3.197309307072032, 3.580972730771657, 4.210936959712628, 4.696233350630486, 5.057787947484691, 5.762856298770622, 6.202442258826556, 7.201777033125233, 7.494788827452983, 8.068876595491768, 8.539671047775760, 8.832141531482335, 9.378046536614405, 10.19737482589787, 10.72974170717674, 11.08493872019481, 11.45180113563216, 12.40504438064274, 12.67278234580962, 13.01015179100566, 13.45596189468759, 13.93802108703619

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