Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19 $
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 + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s − 13-s + 14-s − 15-s + 16-s + 2·17-s − 18-s − 19-s + 20-s + 21-s + 4·22-s + 24-s + 25-s + 26-s − 27-s − 28-s + 10·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.85·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(51870\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{51870} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 51870,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.221973230\)
\(L(\frac12)\)  \(\approx\)  \(1.221973230\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 + T \)
19 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−14.50574092332470, −13.96840812235069, −13.32266563065799, −12.86587087230096, −12.47693607386465, −11.71546896878432, −11.50203356505700, −10.55458989703777, −10.40250095353669, −9.865515146653148, −9.527470363845311, −8.674983112463281, −8.175251456156253, −7.740425745780187, −7.053088325052666, −6.498668550783741, −5.996255148756861, −5.530694209129765, −4.700467071527245, −4.394079008296407, −3.103648940322852, −2.808030260538127, −2.085238726333381, −1.112630158498510, −0.5107329057695889, 0.5107329057695889, 1.112630158498510, 2.085238726333381, 2.808030260538127, 3.103648940322852, 4.394079008296407, 4.700467071527245, 5.530694209129765, 5.996255148756861, 6.498668550783741, 7.053088325052666, 7.740425745780187, 8.175251456156253, 8.674983112463281, 9.527470363845311, 9.865515146653148, 10.40250095353669, 10.55458989703777, 11.50203356505700, 11.71546896878432, 12.47693607386465, 12.86587087230096, 13.32266563065799, 13.96840812235069, 14.50574092332470

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