Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 67 $
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 − 2·4-s − 5-s + 2·7-s + 9-s − 6·11-s − 2·12-s + 2·13-s − 15-s + 4·16-s − 3·17-s − 19-s + 2·20-s + 2·21-s − 9·23-s + 25-s + 27-s − 4·28-s + 3·29-s − 4·31-s − 6·33-s − 2·35-s − 2·36-s − 7·37-s + 2·39-s + 6·41-s − 10·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.577·12-s + 0.554·13-s − 0.258·15-s + 16-s − 0.727·17-s − 0.229·19-s + 0.447·20-s + 0.436·21-s − 1.87·23-s + 1/5·25-s + 0.192·27-s − 0.755·28-s + 0.557·29-s − 0.718·31-s − 1.04·33-s − 0.338·35-s − 1/3·36-s − 1.15·37-s + 0.320·39-s + 0.937·41-s − 1.52·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1005\)    =    \(3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1005} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1005,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;5,\;67\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
5 \( 1 + T \)
67 \( 1 - T \)
good2 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 15 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−19.93619010744192, −18.92757649529904, −18.42216250525735, −17.95333409675382, −17.29915022650742, −15.98304492365033, −15.73001297416115, −14.80582357013523, −14.06478995301481, −13.52989043055441, −12.83272473252637, −12.09756711646390, −10.92241240579635, −10.39223249344379, −9.469806900331238, −8.387459012074038, −8.214779397591055, −7.406186244720636, −5.950607527960282, −4.959416910530016, −4.284879197561755, −3.244943542517303, −1.924096822404054, 0, 1.924096822404054, 3.244943542517303, 4.284879197561755, 4.959416910530016, 5.950607527960282, 7.406186244720636, 8.214779397591055, 8.387459012074038, 9.469806900331238, 10.39223249344379, 10.92241240579635, 12.09756711646390, 12.83272473252637, 13.52989043055441, 14.06478995301481, 14.80582357013523, 15.73001297416115, 15.98304492365033, 17.29915022650742, 17.95333409675382, 18.42216250525735, 18.92757649529904, 19.93619010744192

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