# Properties

 Degree 2 Conductor $2 \cdot 5^{2}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + 3-s + 4-s − 6-s + 2·7-s − 8-s − 2·9-s − 3·11-s + 12-s − 4·13-s − 2·14-s + 16-s − 3·17-s + 2·18-s + 5·19-s + 2·21-s + 3·22-s + 6·23-s − 24-s + 4·26-s − 5·27-s + 2·28-s + 2·31-s − 32-s − 3·33-s + 3·34-s − 2·36-s + ⋯
 L(s)  = 1 − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.904·11-s + 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 1.14·19-s + 0.436·21-s + 0.639·22-s + 1.25·23-s − 0.204·24-s + 0.784·26-s − 0.962·27-s + 0.377·28-s + 0.359·31-s − 0.176·32-s − 0.522·33-s + 0.514·34-s − 1/3·36-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$50$$    =    $$2 \cdot 5^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{50} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 50,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.7131649814$ $L(\frac12)$ $\approx$ $0.7131649814$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;5\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + T$$
5 $$1$$
good3 $$1 - T + p T^{2}$$
7 $$1 - 2 T + p T^{2}$$
11 $$1 + 3 T + p T^{2}$$
13 $$1 + 4 T + p T^{2}$$
17 $$1 + 3 T + p T^{2}$$
19 $$1 - 5 T + p T^{2}$$
23 $$1 - 6 T + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 - 2 T + p T^{2}$$
37 $$1 - 2 T + p T^{2}$$
41 $$1 + 3 T + p T^{2}$$
43 $$1 + 4 T + p T^{2}$$
47 $$1 - 12 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 + 13 T + p T^{2}$$
71 $$1 - 12 T + p T^{2}$$
73 $$1 - 11 T + p T^{2}$$
79 $$1 + 10 T + p T^{2}$$
83 $$1 + 9 T + p T^{2}$$
89 $$1 - 15 T + p T^{2}$$
97 $$1 - 2 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

−19.85985804066716, −18.75556473530924, −17.73623770272490, −16.90369970531125, −15.49987605150771, −14.64380916664077, −13.46639701582307, −11.89949732310571, −10.81431448769284, −9.447991916742082, −8.316812877075225, −7.298531347937641, −5.205921487885761, −2.651550404172352, 2.651550404172352, 5.205921487885761, 7.298531347937641, 8.316812877075225, 9.447991916742082, 10.81431448769284, 11.89949732310571, 13.46639701582307, 14.64380916664077, 15.49987605150771, 16.90369970531125, 17.73623770272490, 18.75556473530924, 19.85985804066716