# Properties

 Degree 2 Conductor $2^{6} \cdot 5$ Sign $1$ Motivic weight 5 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 6·3-s + 25·5-s − 118·7-s − 207·9-s − 192·11-s − 1.10e3·13-s − 150·15-s + 762·17-s + 2.74e3·19-s + 708·21-s + 1.56e3·23-s + 625·25-s + 2.70e3·27-s − 5.91e3·29-s − 6.86e3·31-s + 1.15e3·33-s − 2.95e3·35-s + 5.51e3·37-s + 6.63e3·39-s − 378·41-s + 2.43e3·43-s − 5.17e3·45-s + 1.31e4·47-s − 2.88e3·49-s − 4.57e3·51-s + 9.17e3·53-s − 4.80e3·55-s + ⋯
 L(s)  = 1 − 0.384·3-s + 0.447·5-s − 0.910·7-s − 0.851·9-s − 0.478·11-s − 1.81·13-s − 0.172·15-s + 0.639·17-s + 1.74·19-s + 0.350·21-s + 0.617·23-s + 1/5·25-s + 0.712·27-s − 1.30·29-s − 1.28·31-s + 0.184·33-s − 0.407·35-s + 0.662·37-s + 0.698·39-s − 0.0351·41-s + 0.200·43-s − 0.380·45-s + 0.866·47-s − 0.171·49-s − 0.246·51-s + 0.448·53-s − 0.213·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$320$$    =    $$2^{6} \cdot 5$$ $$\varepsilon$$ = $1$ motivic weight = $$5$$ character : $\chi_{320} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(2,\ 320,\ (\ :5/2),\ 1)$$ $$L(3)$$ $$\approx$$ $$1.085041162$$ $$L(\frac12)$$ $$\approx$$ $$1.085041162$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;5\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
5 $$1 - p^{2} T$$
good3 $$1 + 2 p T + p^{5} T^{2}$$
7 $$1 + 118 T + p^{5} T^{2}$$
11 $$1 + 192 T + p^{5} T^{2}$$
13 $$1 + 1106 T + p^{5} T^{2}$$
17 $$1 - 762 T + p^{5} T^{2}$$
19 $$1 - 2740 T + p^{5} T^{2}$$
23 $$1 - 1566 T + p^{5} T^{2}$$
29 $$1 + 5910 T + p^{5} T^{2}$$
31 $$1 + 6868 T + p^{5} T^{2}$$
37 $$1 - 5518 T + p^{5} T^{2}$$
41 $$1 + 378 T + p^{5} T^{2}$$
43 $$1 - 2434 T + p^{5} T^{2}$$
47 $$1 - 13122 T + p^{5} T^{2}$$
53 $$1 - 9174 T + p^{5} T^{2}$$
59 $$1 - 34980 T + p^{5} T^{2}$$
61 $$1 - 9838 T + p^{5} T^{2}$$
67 $$1 + 33722 T + p^{5} T^{2}$$
71 $$1 - 70212 T + p^{5} T^{2}$$
73 $$1 - 21986 T + p^{5} T^{2}$$
79 $$1 - 4520 T + p^{5} T^{2}$$
83 $$1 - 109074 T + p^{5} T^{2}$$
89 $$1 - 38490 T + p^{5} T^{2}$$
97 $$1 + 1918 T + p^{5} 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

−10.77123803035028232960763292825, −9.692436294261665437103490491608, −9.313143357024145647337940734346, −7.74076691050945123801529012318, −6.97379482944925067542154875019, −5.62036463732788811931789966212, −5.19754526426276643341124078864, −3.36475918352649418176800414997, −2.41797532320575322759885164950, −0.56142460750480993701946855652, 0.56142460750480993701946855652, 2.41797532320575322759885164950, 3.36475918352649418176800414997, 5.19754526426276643341124078864, 5.62036463732788811931789966212, 6.97379482944925067542154875019, 7.74076691050945123801529012318, 9.313143357024145647337940734346, 9.692436294261665437103490491608, 10.77123803035028232960763292825