Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
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 + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s − 2·13-s + 14-s + 16-s + 6·17-s − 4·19-s − 20-s − 22-s + 25-s + 2·26-s − 28-s + 2·29-s − 8·31-s − 32-s − 6·34-s + 35-s + 6·37-s + 4·38-s + 40-s + 6·41-s − 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s + 0.986·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6930\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6930} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6930,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−17.39278135920660, −16.86928596894692, −16.41622152665326, −15.96670250734058, −15.10609830189476, −14.63889899453464, −14.24279919257912, −13.08490510782446, −12.70431351643323, −12.02902786857183, −11.48713624591514, −10.79981887033424, −10.12962469580444, −9.627646799217011, −8.971458352490915, −8.256871345242795, −7.646234912552796, −7.101683878569669, −6.315717129117722, −5.652893432125752, −4.739031046755036, −3.822205369003931, −3.125734592393521, −2.195218997580265, −1.123375331480933, 0, 1.123375331480933, 2.195218997580265, 3.125734592393521, 3.822205369003931, 4.739031046755036, 5.652893432125752, 6.315717129117722, 7.101683878569669, 7.646234912552796, 8.256871345242795, 8.971458352490915, 9.627646799217011, 10.12962469580444, 10.79981887033424, 11.48713624591514, 12.02902786857183, 12.70431351643323, 13.08490510782446, 14.24279919257912, 14.63889899453464, 15.10609830189476, 15.96670250734058, 16.41622152665326, 16.86928596894692, 17.39278135920660

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