Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17 $
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 + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s − 2·11-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s − 18-s + 20-s − 21-s + 2·22-s + 2·23-s + 24-s + 25-s + 26-s − 27-s + 28-s + 2·29-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 − 0.603·11-s − 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(46410\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{46410} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 46410,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13,\;17\}$, 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 \( 1 + T \)
17 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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

−15.13997122805430, −14.24304745394468, −13.93996224846445, −13.21479849070401, −12.78566184329561, −12.15113635597980, −11.67606865233874, −11.22770212487396, −10.50199196325726, −10.31459914019926, −9.745121985340762, −9.135268146663885, −8.495076693215368, −8.122673286516867, −7.376770592884588, −6.903741459509743, −6.386916426180262, −5.681394064729086, −5.221414161430397, −4.619707543443378, −3.920456949123990, −2.872396640111853, −2.467132293527191, −1.588791665874328, −0.9303916856048834, 0, 0.9303916856048834, 1.588791665874328, 2.467132293527191, 2.872396640111853, 3.920456949123990, 4.619707543443378, 5.221414161430397, 5.681394064729086, 6.386916426180262, 6.903741459509743, 7.376770592884588, 8.122673286516867, 8.495076693215368, 9.135268146663885, 9.745121985340762, 10.31459914019926, 10.50199196325726, 11.22770212487396, 11.67606865233874, 12.15113635597980, 12.78566184329561, 13.21479849070401, 13.93996224846445, 14.24304745394468, 15.13997122805430

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