Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 23 $
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 + 12-s + 13-s + 14-s + 15-s + 16-s + 6·17-s + 18-s − 4·19-s + 20-s + 21-s − 23-s + 24-s + 25-s + 26-s + 27-s + 28-s + 6·29-s + 30-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 + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 62790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 62790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(62790\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 23\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{62790} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 62790,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.476931880$
$L(\frac12)$  $\approx$  $7.476931880$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−14.21372276886191, −13.81548229664217, −13.41717340037943, −12.77475933261358, −12.31038144082934, −11.93765869449515, −11.28836054200685, −10.65080195060288, −10.18227338304998, −9.832406715358327, −9.105546831054648, −8.414073069329631, −8.136931507192695, −7.528749538518651, −6.869753647436478, −6.288612724238941, −5.849274404254392, −5.189955035134022, −4.548177794765086, −4.145840993369331, −3.313418922826820, −2.863692053977605, −2.208771050377018, −1.486102267368124, −0.8262231193237672, 0.8262231193237672, 1.486102267368124, 2.208771050377018, 2.863692053977605, 3.313418922826820, 4.145840993369331, 4.548177794765086, 5.189955035134022, 5.849274404254392, 6.288612724238941, 6.869753647436478, 7.528749538518651, 8.136931507192695, 8.414073069329631, 9.105546831054648, 9.832406715358327, 10.18227338304998, 10.65080195060288, 11.28836054200685, 11.93765869449515, 12.31038144082934, 12.77475933261358, 13.41717340037943, 13.81548229664217, 14.21372276886191

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