# Properties

 Degree 2 Conductor 37 Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

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## Dirichlet series

 L(s)  = 1 − 2·2-s − 3·3-s + 2·4-s − 2·5-s + 6·6-s − 7-s + 6·9-s + 4·10-s − 5·11-s − 6·12-s − 2·13-s + 2·14-s + 6·15-s − 4·16-s − 12·18-s − 4·20-s + 3·21-s + 10·22-s + 2·23-s − 25-s + 4·26-s − 9·27-s − 2·28-s + 6·29-s − 12·30-s − 4·31-s + 8·32-s + ⋯
 L(s)  = 1 − 1.41·2-s − 1.73·3-s + 4-s − 0.894·5-s + 2.44·6-s − 0.377·7-s + 2·9-s + 1.26·10-s − 1.50·11-s − 1.73·12-s − 0.554·13-s + 0.534·14-s + 1.54·15-s − 16-s − 2.82·18-s − 0.894·20-s + 0.654·21-s + 2.13·22-s + 0.417·23-s − 1/5·25-s + 0.784·26-s − 1.73·27-s − 0.377·28-s + 1.11·29-s − 2.19·30-s − 0.718·31-s + 1.41·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$37$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{37} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 37,\ (\ :1/2),\ -1)$$ $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 37$,$F_p(T) = 1 - a_p T + p T^2 .$If $p = 37$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad37 $$1 + T$$
good2 $$1 + p T + p T^{2}$$
3 $$1 + p T + p T^{2}$$
5 $$1 + 2 T + p T^{2}$$
7 $$1 + T + p T^{2}$$
11 $$1 + 5 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 - 2 T + p T^{2}$$
29 $$1 - 6 T + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
41 $$1 + 9 T + p T^{2}$$
43 $$1 - 2 T + p T^{2}$$
47 $$1 + 9 T + p T^{2}$$
53 $$1 - T + p T^{2}$$
59 $$1 - 8 T + p T^{2}$$
61 $$1 + 8 T + p T^{2}$$
67 $$1 - 8 T + p T^{2}$$
71 $$1 - 9 T + p T^{2}$$
73 $$1 + T + p T^{2}$$
79 $$1 - 4 T + p T^{2}$$
83 $$1 + 15 T + p T^{2}$$
89 $$1 - 4 T + p T^{2}$$
97 $$1 - 4 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

−19.81482224536338, −18.78719562466392, −18.06365420291071, −17.14169364801487, −16.19201741687448, −15.60385787320432, −12.95838641388285, −11.75732472284978, −10.77513816254080, −9.933098353605352, −8.014330807872879, −6.870391216954432, −5.003170014006659, 0, 5.003170014006659, 6.870391216954432, 8.014330807872879, 9.933098353605352, 10.77513816254080, 11.75732472284978, 12.95838641388285, 15.60385787320432, 16.19201741687448, 17.14169364801487, 18.06365420291071, 18.78719562466392, 19.81482224536338