# Properties

 Degree 2 Conductor 43 Sign $1$ Motivic weight 3 Primitive yes Self-dual yes Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − 3.31·2-s − 7.10·3-s + 2.98·4-s + 8.51·5-s + 23.5·6-s + 9.84·7-s + 16.6·8-s + 23.4·9-s − 28.2·10-s + 0.597·11-s − 21.2·12-s + 34.9·13-s − 32.6·14-s − 60.5·15-s − 78.9·16-s + 91.3·17-s − 77.7·18-s − 24.0·19-s + 25.4·20-s − 69.9·21-s − 1.98·22-s + 127.·23-s − 118.·24-s − 52.4·25-s − 115.·26-s + 25.0·27-s + 29.3·28-s + ⋯
 L(s)  = 1 − 1.17·2-s − 1.36·3-s + 0.373·4-s + 0.761·5-s + 1.60·6-s + 0.531·7-s + 0.734·8-s + 0.869·9-s − 0.892·10-s + 0.0163·11-s − 0.510·12-s + 0.744·13-s − 0.622·14-s − 1.04·15-s − 1.23·16-s + 1.30·17-s − 1.01·18-s − 0.290·19-s + 0.284·20-s − 0.726·21-s − 0.0192·22-s + 1.15·23-s − 1.00·24-s − 0.419·25-s − 0.872·26-s + 0.178·27-s + 0.198·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $1$ motivic weight = $$3$$ character : $\chi_{43} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :3/2),\ 1)$$ $$L(2)$$ $$\approx$$ $$0.552289$$ $$L(\frac12)$$ $$\approx$$ $$0.552289$$ $$L(\frac{5}{2})$$ 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 + 43T$$
good2 $$1 + 3.31T + 8T^{2}$$
3 $$1 + 7.10T + 27T^{2}$$
5 $$1 - 8.51T + 125T^{2}$$
7 $$1 - 9.84T + 343T^{2}$$
11 $$1 - 0.597T + 1.33e3T^{2}$$
13 $$1 - 34.9T + 2.19e3T^{2}$$
17 $$1 - 91.3T + 4.91e3T^{2}$$
19 $$1 + 24.0T + 6.85e3T^{2}$$
23 $$1 - 127.T + 1.21e4T^{2}$$
29 $$1 - 285.T + 2.43e4T^{2}$$
31 $$1 + 192.T + 2.97e4T^{2}$$
37 $$1 - 363.T + 5.06e4T^{2}$$
41 $$1 - 191.T + 6.89e4T^{2}$$
47 $$1 + 458.T + 1.03e5T^{2}$$
53 $$1 + 583.T + 1.48e5T^{2}$$
59 $$1 - 416.T + 2.05e5T^{2}$$
61 $$1 + 30.9T + 2.26e5T^{2}$$
67 $$1 - 16.1T + 3.00e5T^{2}$$
71 $$1 + 219.T + 3.57e5T^{2}$$
73 $$1 - 1.03e3T + 3.89e5T^{2}$$
79 $$1 - 445.T + 4.93e5T^{2}$$
83 $$1 - 298.T + 5.71e5T^{2}$$
89 $$1 - 846.T + 7.04e5T^{2}$$
97 $$1 + 259.T + 9.12e5T^{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

−16.18029482951591806374371234538, −14.32536331491568879032141790091, −12.92641654582797863343248816646, −11.42633349989367933165246452880, −10.58824249222854177152601334428, −9.518008966316237270164521160893, −8.037654713362921648572062740172, −6.37730194293215393695599291147, −5.02358574249926313807328819262, −1.12975603138289563510018708281, 1.12975603138289563510018708281, 5.02358574249926313807328819262, 6.37730194293215393695599291147, 8.037654713362921648572062740172, 9.518008966316237270164521160893, 10.58824249222854177152601334428, 11.42633349989367933165246452880, 12.92641654582797863343248816646, 14.32536331491568879032141790091, 16.18029482951591806374371234538