Properties

Degree 2
Conductor 43
Sign $0.731 - 0.681i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 7.49i·2-s + 14.5i·3-s − 40.1·4-s + 9.40i·5-s + 108.·6-s + 53.5i·7-s + 180. i·8-s − 129.·9-s + 70.4·10-s + 151.·11-s − 582. i·12-s − 319.·13-s + 401.·14-s − 136.·15-s + 713.·16-s + 32.2·17-s + ⋯
L(s)  = 1  − 1.87i·2-s + 1.61i·3-s − 2.50·4-s + 0.376i·5-s + 3.02·6-s + 1.09i·7-s + 2.82i·8-s − 1.60·9-s + 0.704·10-s + 1.24·11-s − 4.04i·12-s − 1.89·13-s + 2.04·14-s − 0.606·15-s + 2.78·16-s + 0.111·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $0.731 - 0.681i$
motivic weight  =  \(4\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :2),\ 0.731 - 0.681i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.910207 + 0.358450i\)
\(L(\frac12)\)  \(\approx\)  \(0.910207 + 0.358450i\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 2. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad43 \( 1 + (-1.35e3 + 1.26e3i)T \)
good2 \( 1 + 7.49iT - 16T^{2} \)
3 \( 1 - 14.5iT - 81T^{2} \)
5 \( 1 - 9.40iT - 625T^{2} \)
7 \( 1 - 53.5iT - 2.40e3T^{2} \)
11 \( 1 - 151.T + 1.46e4T^{2} \)
13 \( 1 + 319.T + 2.85e4T^{2} \)
17 \( 1 - 32.2T + 8.35e4T^{2} \)
19 \( 1 - 304. iT - 1.30e5T^{2} \)
23 \( 1 + 199.T + 2.79e5T^{2} \)
29 \( 1 - 268. iT - 7.07e5T^{2} \)
31 \( 1 - 665.T + 9.23e5T^{2} \)
37 \( 1 + 1.17e3iT - 1.87e6T^{2} \)
41 \( 1 - 212.T + 2.82e6T^{2} \)
47 \( 1 + 3.10e3T + 4.87e6T^{2} \)
53 \( 1 - 2.66e3T + 7.89e6T^{2} \)
59 \( 1 - 2.74e3T + 1.21e7T^{2} \)
61 \( 1 - 5.88e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.09e3T + 2.01e7T^{2} \)
71 \( 1 - 5.84e3iT - 2.54e7T^{2} \)
73 \( 1 + 663. iT - 2.83e7T^{2} \)
79 \( 1 - 6.32e3T + 3.89e7T^{2} \)
83 \( 1 - 8.35e3T + 4.74e7T^{2} \)
89 \( 1 - 4.25e3iT - 6.27e7T^{2} \)
97 \( 1 + 3.58e3T + 8.85e7T^{2} \)
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

−14.85573042810230700479955601759, −14.38965795599064119859250067100, −12.32418339133577563517125872777, −11.69779987955800989150351380729, −10.42245338230509279328935417844, −9.642892147151646322229815737282, −8.856047613267616014365078790571, −5.21180956795999054692394872920, −3.96391021400871232148927834228, −2.58297413036754763427615799148, 0.66081432769580188627961834943, 4.72251665884262132833205219136, 6.52546708419537694418509711302, 7.15324638796110571059241292902, 8.095012586789821372641335223194, 9.514956026945666949651817206898, 12.11188949055336962393846516883, 13.19285402444284560807649657223, 14.05119709530819903126455625498, 14.81761843105858915231702981074

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