# Properties

 Degree 2 Conductor 73 Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 2-s − 4-s + 2·5-s + 2·7-s − 3·8-s − 3·9-s + 2·10-s − 2·11-s − 6·13-s + 2·14-s − 16-s + 2·17-s − 3·18-s + 8·19-s − 2·20-s − 2·22-s + 4·23-s − 25-s − 6·26-s − 2·28-s + 2·29-s − 2·31-s + 5·32-s + 2·34-s + 4·35-s + 3·36-s − 6·37-s + ⋯
 L(s)  = 1 + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.755·7-s − 1.06·8-s − 9-s + 0.632·10-s − 0.603·11-s − 1.66·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s − 0.707·18-s + 1.83·19-s − 0.447·20-s − 0.426·22-s + 0.834·23-s − 1/5·25-s − 1.17·26-s − 0.377·28-s + 0.371·29-s − 0.359·31-s + 0.883·32-s + 0.342·34-s + 0.676·35-s + 1/2·36-s − 0.986·37-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$73$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{73} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 73,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $1.182660467$ $L(\frac12)$ $\approx$ $1.182660467$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \neq 73$, $F_p(T) = 1 - a_p T + p T^2 .$If $p = 73$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad73 $$1 - T$$
good2 $$1 - T + p T^{2}$$
3 $$1 + p T^{2}$$
5 $$1 - 2 T + p T^{2}$$
7 $$1 - 2 T + p T^{2}$$
11 $$1 + 2 T + p T^{2}$$
13 $$1 + 6 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 - 8 T + p T^{2}$$
23 $$1 - 4 T + p T^{2}$$
29 $$1 - 2 T + p T^{2}$$
31 $$1 + 2 T + p T^{2}$$
37 $$1 + 6 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 + 2 T + p T^{2}$$
47 $$1 - 6 T + p T^{2}$$
53 $$1 - 10 T + p T^{2}$$
59 $$1 + 6 T + p T^{2}$$
61 $$1 + 14 T + p T^{2}$$
67 $$1 - 8 T + p T^{2}$$
71 $$1 + p T^{2}$$
79 $$1 + 4 T + p T^{2}$$
83 $$1 + 14 T + p T^{2}$$
89 $$1 + 6 T + p T^{2}$$
97 $$1 + 10 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

−19.78071511778486, −18.37843267445360, −17.63067364922466, −16.93644286863809, −15.25152910789263, −14.20095311456822, −13.90079370363007, −12.55583978793064, −11.60505678784584, −10.02750085781247, −9.047241577107533, −7.591744378983860, −5.588364505746148, −5.030533566858957, −2.851725941271704, 2.851725941271704, 5.030533566858957, 5.588364505746148, 7.591744378983860, 9.047241577107533, 10.02750085781247, 11.60505678784584, 12.55583978793064, 13.90079370363007, 14.20095311456822, 15.25152910789263, 16.93644286863809, 17.63067364922466, 18.37843267445360, 19.78071511778486