Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \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 + 11-s + 12-s + 2·13-s + 14-s − 15-s + 16-s + 6·17-s + 18-s + 19-s − 20-s + 21-s + 22-s + 24-s + 25-s + 2·26-s + 27-s + 28-s + 6·29-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 + 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 43890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 43890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43890\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{43890} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 43890,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.205830805$
$L(\frac12)$  $\approx$  $6.205830805$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;11,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 - T \)
19 \( 1 - T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−14.46209894516634, −14.35992085997327, −13.65321439050832, −13.29547654428961, −12.44555712588268, −12.29477344955178, −11.65643944624380, −11.14583483209207, −10.55818426182842, −10.04380294191811, −9.395318437721913, −8.781487237168111, −8.214739051636566, −7.687162258589681, −7.314479658932340, −6.554426400978080, −5.970439005787710, −5.368692816738949, −4.694086148404337, −4.158260385600987, −3.462385340230230, −3.132136135908609, −2.294268583859572, −1.456892353633857, −0.8271498749453087, 0.8271498749453087, 1.456892353633857, 2.294268583859572, 3.132136135908609, 3.462385340230230, 4.158260385600987, 4.694086148404337, 5.368692816738949, 5.970439005787710, 6.554426400978080, 7.314479658932340, 7.687162258589681, 8.214739051636566, 8.781487237168111, 9.395318437721913, 10.04380294191811, 10.55818426182842, 11.14583483209207, 11.65643944624380, 12.29477344955178, 12.44555712588268, 13.29547654428961, 13.65321439050832, 14.35992085997327, 14.46209894516634

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