# Properties

 Degree 2 Conductor 83 Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

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## Dirichlet series

 L(s)  = 1 − 2-s − 3-s − 4-s − 2·5-s + 6-s − 3·7-s + 3·8-s − 2·9-s + 2·10-s + 3·11-s + 12-s − 6·13-s + 3·14-s + 2·15-s − 16-s + 5·17-s + 2·18-s + 2·19-s + 2·20-s + 3·21-s − 3·22-s − 4·23-s − 3·24-s − 25-s + 6·26-s + 5·27-s + 3·28-s + ⋯
 L(s)  = 1 − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s − 1.13·7-s + 1.06·8-s − 2/3·9-s + 0.632·10-s + 0.904·11-s + 0.288·12-s − 1.66·13-s + 0.801·14-s + 0.516·15-s − 1/4·16-s + 1.21·17-s + 0.471·18-s + 0.458·19-s + 0.447·20-s + 0.654·21-s − 0.639·22-s − 0.834·23-s − 0.612·24-s − 1/5·25-s + 1.17·26-s + 0.962·27-s + 0.566·28-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$83$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{83} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 83,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 83$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 83$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad83 $$1 + T$$
good2 $$1 + T + p T^{2}$$
3 $$1 + T + p T^{2}$$
5 $$1 + 2 T + p T^{2}$$
7 $$1 + 3 T + p T^{2}$$
11 $$1 - 3 T + p T^{2}$$
13 $$1 + 6 T + p T^{2}$$
17 $$1 - 5 T + p T^{2}$$
19 $$1 - 2 T + p T^{2}$$
23 $$1 + 4 T + p T^{2}$$
29 $$1 + 7 T + p T^{2}$$
31 $$1 - 5 T + p T^{2}$$
37 $$1 + 11 T + p T^{2}$$
41 $$1 + 2 T + p T^{2}$$
43 $$1 + 8 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 - 5 T + p T^{2}$$
61 $$1 - 5 T + p T^{2}$$
67 $$1 + 2 T + p T^{2}$$
71 $$1 - 2 T + p T^{2}$$
73 $$1 + p T^{2}$$
79 $$1 - 14 T + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 + 8 T + p T^{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

−19.42783350289885, −18.98433131920491, −17.62714815703440, −16.85569317903090, −16.30410363710809, −14.86720481310773, −13.82349410001976, −12.30534590659786, −11.77890434965261, −10.17676635448238, −9.457827669916619, −8.118927993931558, −6.974502993044303, −5.313270241803154, −3.654759012444532, 0, 3.654759012444532, 5.313270241803154, 6.974502993044303, 8.118927993931558, 9.457827669916619, 10.17676635448238, 11.77890434965261, 12.30534590659786, 13.82349410001976, 14.86720481310773, 16.30410363710809, 16.85569317903090, 17.62714815703440, 18.98433131920491, 19.42783350289885