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 − 4·11-s − 12-s + 13-s − 14-s − 15-s + 16-s + 17-s − 18-s − 4·19-s + 20-s − 21-s + 4·22-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 − 1.20·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-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 − 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 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−15.02325553177755, −14.46104258602970, −13.73731347003461, −13.25479463203012, −12.75735537706616, −12.28903867492222, −11.62128035636372, −11.00501166059324, −10.82340169692656, −10.09210860104646, −9.870771687152018, −9.099057502755854, −8.513711711790021, −7.994373515671489, −7.588600501134659, −6.764704551234668, −6.406994333893026, −5.692575505801077, −5.207222558321284, −4.684937652882652, −3.813035531183039, −3.050419394067061, −2.236369368634410, −1.769457837924439, −0.8412078634741051, 0, 0.8412078634741051, 1.769457837924439, 2.236369368634410, 3.050419394067061, 3.813035531183039, 4.684937652882652, 5.207222558321284, 5.692575505801077, 6.406994333893026, 6.764704551234668, 7.588600501134659, 7.994373515671489, 8.513711711790021, 9.099057502755854, 9.870771687152018, 10.09210860104646, 10.82340169692656, 11.00501166059324, 11.62128035636372, 12.28903867492222, 12.75735537706616, 13.25479463203012, 13.73731347003461, 14.46104258602970, 15.02325553177755

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