Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 167 $
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 − 4·7-s − 8-s + 9-s − 2·10-s − 4·11-s + 12-s + 4·14-s + 2·15-s + 16-s − 4·17-s − 18-s − 4·19-s + 2·20-s − 4·21-s + 4·22-s − 4·23-s − 24-s − 25-s + 27-s − 4·28-s − 2·29-s − 2·30-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 − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s + 0.288·12-s + 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s − 0.872·21-s + 0.852·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.755·28-s − 0.371·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 1002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1002\)    =    \(2 \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{1002} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 1002,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
167 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
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

−19.69603737484182, −19.18956842472138, −18.47446992799437, −17.83155003992535, −17.15157994139819, −16.26444013105185, −15.72668956251314, −15.17796141103375, −14.01881897203224, −13.30373406958535, −12.93516506776202, −12.01202472407119, −10.65106743077849, −10.23693043348589, −9.562384110123212, −8.872167065392672, −8.076909988757104, −6.999304925603343, −6.343881390912735, −5.473079258399416, −3.957885621149568, −2.750247610946624, −2.081410004873100, 0, 2.081410004873100, 2.750247610946624, 3.957885621149568, 5.473079258399416, 6.343881390912735, 6.999304925603343, 8.076909988757104, 8.872167065392672, 9.562384110123212, 10.23693043348589, 10.65106743077849, 12.01202472407119, 12.93516506776202, 13.30373406958535, 14.01881897203224, 15.17796141103375, 15.72668956251314, 16.26444013105185, 17.15157994139819, 17.83155003992535, 18.47446992799437, 19.18956842472138, 19.69603737484182

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