# Properties

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

# Learn more about

## Dirichlet series

 L(s)  = 1 − 2·5-s − 3·9-s + 6·13-s + 2·17-s − 25-s − 10·29-s − 2·37-s + 10·41-s + 6·45-s − 7·49-s + 14·53-s − 10·61-s − 12·65-s − 6·73-s + 9·81-s − 4·85-s + 10·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 18·117-s + ⋯
 L(s)  = 1 − 0.894·5-s − 9-s + 1.66·13-s + 0.485·17-s − 1/5·25-s − 1.85·29-s − 0.328·37-s + 1.56·41-s + 0.894·45-s − 49-s + 1.92·53-s − 1.28·61-s − 1.48·65-s − 0.702·73-s + 81-s − 0.433·85-s + 1.05·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.66·117-s + ⋯

## Functional equation

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

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$32$$    =    $$2^{5}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{32} (1, \cdot )$ Sato-Tate : $N(\mathrm{U}(1))$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 32,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $0.6555143885$ $L(\frac12)$ $\approx$ $0.6555143885$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$F_p(T) = 1 - a_p T + p T^2 .$If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + p T^{2}$$
5 $$1 + 2 T + p T^{2}$$
7 $$1 + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 - 6 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + 10 T + p T^{2}$$
31 $$1 + p T^{2}$$
37 $$1 + 2 T + p T^{2}$$
41 $$1 - 10 T + p T^{2}$$
43 $$1 + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 - 14 T + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + 10 T + p T^{2}$$
67 $$1 + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 6 T + p T^{2}$$
79 $$1 + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 - 10 T + p T^{2}$$
97 $$1 - 18 T + p T^{2}$$
show more
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

−19.18872457828344, −18.11836294614509, −16.73856366990992, −15.74882074786535, −14.57652563978276, −13.27687552535143, −11.76661268274493, −10.90769214371221, −8.955386231165229, −7.771994739060971, −5.871464188488337, −3.674782226530865, 3.674782226530865, 5.871464188488337, 7.771994739060971, 8.955386231165229, 10.90769214371221, 11.76661268274493, 13.27687552535143, 14.57652563978276, 15.74882074786535, 16.73856366990992, 18.11836294614509, 19.18872457828344