Properties

 Degree 2 Conductor 43 Sign $1$ Motivic weight 5 Primitive yes Self-dual yes Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 − 1.86·2-s − 14.8·3-s − 28.5·4-s − 42.2·5-s + 27.6·6-s + 202.·7-s + 112.·8-s − 21.8·9-s + 78.5·10-s + 436.·11-s + 424.·12-s − 617.·13-s − 377.·14-s + 628.·15-s + 703.·16-s + 833.·17-s + 40.6·18-s + 1.35e3·19-s + 1.20e3·20-s − 3.01e3·21-s − 812.·22-s − 904.·23-s − 1.67e3·24-s − 1.34e3·25-s + 1.14e3·26-s + 3.93e3·27-s − 5.79e3·28-s + ⋯
 L(s)  = 1 − 0.328·2-s − 0.954·3-s − 0.891·4-s − 0.755·5-s + 0.313·6-s + 1.56·7-s + 0.622·8-s − 0.0898·9-s + 0.248·10-s + 1.08·11-s + 0.850·12-s − 1.01·13-s − 0.514·14-s + 0.720·15-s + 0.687·16-s + 0.699·17-s + 0.0295·18-s + 0.862·19-s + 0.673·20-s − 1.49·21-s − 0.357·22-s − 0.356·23-s − 0.593·24-s − 0.429·25-s + 0.333·26-s + 1.03·27-s − 1.39·28-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$43$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : $\chi_{43} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 43,\ (\ :5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$0.771375$$ $$L(\frac12)$$ $$\approx$$ $$0.771375$$ $$L(\frac{7}{2})$$ 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 - 1.84e3T$$
good2 $$1 + 1.86T + 32T^{2}$$
3 $$1 + 14.8T + 243T^{2}$$
5 $$1 + 42.2T + 3.12e3T^{2}$$
7 $$1 - 202.T + 1.68e4T^{2}$$
11 $$1 - 436.T + 1.61e5T^{2}$$
13 $$1 + 617.T + 3.71e5T^{2}$$
17 $$1 - 833.T + 1.41e6T^{2}$$
19 $$1 - 1.35e3T + 2.47e6T^{2}$$
23 $$1 + 904.T + 6.43e6T^{2}$$
29 $$1 - 5.32e3T + 2.05e7T^{2}$$
31 $$1 - 919.T + 2.86e7T^{2}$$
37 $$1 - 4.96e3T + 6.93e7T^{2}$$
41 $$1 + 5.93e3T + 1.15e8T^{2}$$
47 $$1 - 1.78e4T + 2.29e8T^{2}$$
53 $$1 - 2.47e4T + 4.18e8T^{2}$$
59 $$1 - 3.38e4T + 7.14e8T^{2}$$
61 $$1 - 4.59e4T + 8.44e8T^{2}$$
67 $$1 + 5.85e4T + 1.35e9T^{2}$$
71 $$1 - 1.79e3T + 1.80e9T^{2}$$
73 $$1 + 4.67e4T + 2.07e9T^{2}$$
79 $$1 + 7.94e4T + 3.07e9T^{2}$$
83 $$1 - 9.12e4T + 3.93e9T^{2}$$
89 $$1 + 1.64e4T + 5.58e9T^{2}$$
97 $$1 - 1.45e5T + 8.58e9T^{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

−14.75393250129029423138969816928, −14.04360557292616432918304153379, −12.01930551698968509975794815424, −11.63501157376476368578681355620, −10.16194805356769747799875647556, −8.601699745385936130425124843319, −7.51316870870306800445127543065, −5.38701354559212273686749777793, −4.31147431097332069979659576886, −0.892984695343857725849962469810, 0.892984695343857725849962469810, 4.31147431097332069979659576886, 5.38701354559212273686749777793, 7.51316870870306800445127543065, 8.601699745385936130425124843319, 10.16194805356769747799875647556, 11.63501157376476368578681355620, 12.01930551698968509975794815424, 14.04360557292616432918304153379, 14.75393250129029423138969816928