Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13 \cdot 911 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

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 − 7·17-s − 18-s + 2·19-s − 20-s − 21-s − 8·23-s − 24-s − 4·25-s − 26-s + 27-s − 28-s − 9·29-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 − 1.69·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.218·21-s − 1.66·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.67·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(71058\)    =    \(2 \cdot 3 \cdot 13 \cdot 911\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{71058} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 71058,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;13,\;911\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13,\;911\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
13 \( 1 - T \)
911 \( 1 - T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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

−14.79585473777603, −14.08145307662278, −13.56300584845275, −13.21928580047066, −12.65211836556234, −12.01526624835087, −11.42156387766678, −11.25208270745782, −10.50667473842551, −9.864928255066120, −9.602826528728714, −9.051693631032557, −8.436737654138823, −8.072448036850939, −7.593269574972200, −6.982527709959375, −6.441959648239574, −5.982623882688010, −5.215538029270772, −4.356837525516047, −3.934176669472107, −3.333131939503981, −2.642147485729765, −1.912798142212171, −1.493943783093988, 0, 0, 1.493943783093988, 1.912798142212171, 2.642147485729765, 3.333131939503981, 3.934176669472107, 4.356837525516047, 5.215538029270772, 5.982623882688010, 6.441959648239574, 6.982527709959375, 7.593269574972200, 8.072448036850939, 8.436737654138823, 9.051693631032557, 9.602826528728714, 9.864928255066120, 10.50667473842551, 11.25208270745782, 11.42156387766678, 12.01526624835087, 12.65211836556234, 13.21928580047066, 13.56300584845275, 14.08145307662278, 14.79585473777603

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