Properties

Degree 2
Conductor 43
Sign $-0.996 + 0.0820i$
Motivic weight 10
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 41.3i·2-s − 57.8i·3-s − 688.·4-s + 4.60e3i·5-s + 2.39e3·6-s + 1.65e4i·7-s + 1.38e4i·8-s + 5.57e4·9-s − 1.90e5·10-s + 2.23e5·11-s + 3.98e4i·12-s + 6.52e5·13-s − 6.84e5·14-s + 2.66e5·15-s − 1.27e6·16-s − 5.77e5·17-s + ⋯
L(s)  = 1  + 1.29i·2-s − 0.237i·3-s − 0.672·4-s + 1.47i·5-s + 0.307·6-s + 0.983i·7-s + 0.423i·8-s + 0.943·9-s − 1.90·10-s + 1.38·11-s + 0.160i·12-s + 1.75·13-s − 1.27·14-s + 0.350·15-s − 1.22·16-s − 0.406·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0820i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.996 + 0.0820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(43\)
\( \varepsilon \)  =  $-0.996 + 0.0820i$
motivic weight  =  \(10\)
character  :  $\chi_{43} (42, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 43,\ (\ :5),\ -0.996 + 0.0820i)\)
\(L(\frac{11}{2})\)  \(\approx\)  \(0.101349 - 2.46766i\)
\(L(\frac12)\)  \(\approx\)  \(0.101349 - 2.46766i\)
\(L(6)\)   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 \( 1 + (-1.46e8 + 1.20e7i)T \)
good2 \( 1 - 41.3iT - 1.02e3T^{2} \)
3 \( 1 + 57.8iT - 5.90e4T^{2} \)
5 \( 1 - 4.60e3iT - 9.76e6T^{2} \)
7 \( 1 - 1.65e4iT - 2.82e8T^{2} \)
11 \( 1 - 2.23e5T + 2.59e10T^{2} \)
13 \( 1 - 6.52e5T + 1.37e11T^{2} \)
17 \( 1 + 5.77e5T + 2.01e12T^{2} \)
19 \( 1 - 2.34e6iT - 6.13e12T^{2} \)
23 \( 1 + 4.51e6T + 4.14e13T^{2} \)
29 \( 1 + 9.59e6iT - 4.20e14T^{2} \)
31 \( 1 - 9.38e6T + 8.19e14T^{2} \)
37 \( 1 + 9.20e7iT - 4.80e15T^{2} \)
41 \( 1 + 5.46e7T + 1.34e16T^{2} \)
47 \( 1 + 5.77e7T + 5.25e16T^{2} \)
53 \( 1 + 3.80e8T + 1.74e17T^{2} \)
59 \( 1 - 5.64e8T + 5.11e17T^{2} \)
61 \( 1 + 6.10e8iT - 7.13e17T^{2} \)
67 \( 1 + 1.39e9T + 1.82e18T^{2} \)
71 \( 1 + 2.86e9iT - 3.25e18T^{2} \)
73 \( 1 + 2.78e9iT - 4.29e18T^{2} \)
79 \( 1 - 3.46e9T + 9.46e18T^{2} \)
83 \( 1 - 3.22e9T + 1.55e19T^{2} \)
89 \( 1 + 7.71e8iT - 3.11e19T^{2} \)
97 \( 1 - 9.01e9T + 7.37e19T^{2} \)
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

−14.59310239094410552325727422323, −13.71671387515398501669862747193, −11.94985467905046124063332012015, −10.79728224196455384143383499149, −9.116723368426925148961058628638, −7.79199211563205227145789319893, −6.48797482513042704292224139314, −6.11291507722890807522282984745, −3.81839019441135441119430283683, −1.91693511059875007812831112300, 0.965747897241147854764484274353, 1.37064642788171625998842932140, 3.81664276853104015160209178786, 4.42893907485786235256224888310, 6.68730914618223062577121605459, 8.684872636899361612813938104331, 9.631787654857346744848268962479, 10.84175717440083723110957430354, 11.88632420519190181633275072654, 13.03935963593612351985110215930

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