# Properties

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

# Origins

## Dirichlet series

 L(s)  = 1 − 8·3-s + 14·5-s − 208·7-s − 179·9-s − 536·11-s + 694·13-s − 112·15-s − 1.27e3·17-s + 1.11e3·19-s + 1.66e3·21-s + 3.21e3·23-s − 2.92e3·25-s + 3.37e3·27-s + 2.91e3·29-s − 2.62e3·31-s + 4.28e3·33-s − 2.91e3·35-s − 9.45e3·37-s − 5.55e3·39-s + 170·41-s − 1.99e4·43-s − 2.50e3·45-s + 32·47-s + 2.64e4·49-s + 1.02e4·51-s − 2.21e4·53-s − 7.50e3·55-s + ⋯
 L(s)  = 1 − 0.513·3-s + 0.250·5-s − 1.60·7-s − 0.736·9-s − 1.33·11-s + 1.13·13-s − 0.128·15-s − 1.07·17-s + 0.706·19-s + 0.823·21-s + 1.26·23-s − 0.937·25-s + 0.891·27-s + 0.644·29-s − 0.490·31-s + 0.685·33-s − 0.401·35-s − 1.13·37-s − 0.584·39-s + 0.0157·41-s − 1.64·43-s − 0.184·45-s + 0.00211·47-s + 1.57·49-s + 0.550·51-s − 1.08·53-s − 0.334·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$32$$    =    $$2^{5}$$ $$\varepsilon$$ = $-1$ motivic weight = $$5$$ character : $\chi_{32} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = $$1$$ Selberg data = $$(2,\ 32,\ (\ :5/2),\ -1)$$ $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 8 T + p^{5} T^{2}$$
5 $$1 - 14 T + p^{5} T^{2}$$
7 $$1 + 208 T + p^{5} T^{2}$$
11 $$1 + 536 T + p^{5} T^{2}$$
13 $$1 - 694 T + p^{5} T^{2}$$
17 $$1 + 1278 T + p^{5} T^{2}$$
19 $$1 - 1112 T + p^{5} T^{2}$$
23 $$1 - 3216 T + p^{5} T^{2}$$
29 $$1 - 2918 T + p^{5} T^{2}$$
31 $$1 + 2624 T + p^{5} T^{2}$$
37 $$1 + 9458 T + p^{5} T^{2}$$
41 $$1 - 170 T + p^{5} T^{2}$$
43 $$1 + 19928 T + p^{5} T^{2}$$
47 $$1 - 32 T + p^{5} T^{2}$$
53 $$1 + 22178 T + p^{5} T^{2}$$
59 $$1 - 41480 T + p^{5} T^{2}$$
61 $$1 - 15462 T + p^{5} T^{2}$$
67 $$1 + 20744 T + p^{5} T^{2}$$
71 $$1 - 28592 T + p^{5} T^{2}$$
73 $$1 + 53670 T + p^{5} T^{2}$$
79 $$1 + 69152 T + p^{5} T^{2}$$
83 $$1 + 37800 T + p^{5} T^{2}$$
89 $$1 + 126806 T + p^{5} T^{2}$$
97 $$1 - 62290 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

−15.58207076215561775169825543226, −13.61336521132877859098282974396, −12.90187517906477616610015363559, −11.31857003666304005985439474032, −10.14003899981572732495857398518, −8.710965820746736160627417431972, −6.70672702287718643748508040768, −5.50818870005420743787505703747, −3.08685227262836591523781351559, 0, 3.08685227262836591523781351559, 5.50818870005420743787505703747, 6.70672702287718643748508040768, 8.710965820746736160627417431972, 10.14003899981572732495857398518, 11.31857003666304005985439474032, 12.90187517906477616610015363559, 13.61336521132877859098282974396, 15.58207076215561775169825543226