# Properties

 Degree 2 Conductor $2^{3} \cdot 5 \cdot 29^{2}$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 5-s − 4·7-s − 3·9-s − 4·11-s − 2·13-s − 2·17-s − 4·19-s + 4·23-s + 25-s + 8·31-s − 4·35-s − 6·37-s + 6·41-s + 8·43-s − 3·45-s − 4·47-s + 9·49-s + 6·53-s − 4·55-s − 4·59-s + 2·61-s + 12·63-s − 2·65-s + 8·67-s + 6·73-s + 16·77-s + 9·81-s + ⋯
 L(s)  = 1 + 0.447·5-s − 1.51·7-s − 9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.248·65-s + 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$33640$$    =    $$2^{3} \cdot 5 \cdot 29^{2}$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{33640} (1, \cdot )$ Sato-Tate : $\mathrm{SU}(2)$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 33640,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5,\;29\}$,$F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;5,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
5 $$1 - T$$
29 $$1$$
good3 $$1 + p T^{2}$$
7 $$1 + 4 T + p T^{2}$$
11 $$1 + 4 T + p T^{2}$$
13 $$1 + 2 T + p T^{2}$$
17 $$1 + 2 T + p T^{2}$$
19 $$1 + 4 T + p T^{2}$$
23 $$1 - 4 T + p T^{2}$$
31 $$1 - 8 T + p T^{2}$$
37 $$1 + 6 T + p T^{2}$$
41 $$1 - 6 T + p T^{2}$$
43 $$1 - 8 T + p T^{2}$$
47 $$1 + 4 T + p T^{2}$$
53 $$1 - 6 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 - 2 T + p T^{2}$$
67 $$1 - 8 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 - 6 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + 16 T + p T^{2}$$
89 $$1 - 6 T + p T^{2}$$
97 $$1 - 14 T + p T^{2}$$
show less
\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

−15.37549537310899, −14.74048752760684, −14.12860792490237, −13.60331584853034, −13.15571969785026, −12.62316671210797, −12.37864159403511, −11.47539359568889, −10.96791682530584, −10.29915579329998, −10.08156225923280, −9.293785194713515, −8.911913213139615, −8.318545880791698, −7.646250991061884, −6.928786853465612, −6.430432123091344, −5.940951588519136, −5.322372758807355, −4.747384525950520, −3.904337127808482, −3.046340851169198, −2.664694133115335, −2.181195691877715, −0.7322024083033291, 0, 0.7322024083033291, 2.181195691877715, 2.664694133115335, 3.046340851169198, 3.904337127808482, 4.747384525950520, 5.322372758807355, 5.940951588519136, 6.430432123091344, 6.928786853465612, 7.646250991061884, 8.318545880791698, 8.911913213139615, 9.293785194713515, 10.08156225923280, 10.29915579329998, 10.96791682530584, 11.47539359568889, 12.37864159403511, 12.62316671210797, 13.15571969785026, 13.60331584853034, 14.12860792490237, 14.74048752760684, 15.37549537310899