# Properties

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

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## 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}$$
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\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

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