# Properties

 Degree 4 Conductor $2^{8} \cdot 3^{4}$ Sign $1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 − 2·4-s + 2·7-s + 2·13-s + 4·16-s + 2·25-s − 4·28-s + 2·31-s + 14·37-s + 2·49-s − 4·52-s − 10·61-s − 8·64-s + 16·67-s − 22·79-s + 4·91-s − 4·100-s + 101-s + 103-s + 107-s + 109-s + 8·112-s + 113-s + 2·121-s − 4·124-s + 127-s + 131-s + 137-s + ⋯
 L(s)  = 1 − 4-s + 0.755·7-s + 0.554·13-s + 16-s + 2/5·25-s − 0.755·28-s + 0.359·31-s + 2.30·37-s + 2/7·49-s − 0.554·52-s − 1.28·61-s − 64-s + 1.95·67-s − 2.47·79-s + 0.419·91-s − 2/5·100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.755·112-s + 0.0940·113-s + 2/11·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$20736$$    =    $$2^{8} \cdot 3^{4}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{20736} (1, \cdot )$ Sato-Tate : $J(E_4)$ primitive : yes self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 20736,\ (\ :1/2, 1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$1.033342181$$ $$L(\frac12)$$ $$\approx$$ $$1.033342181$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ $$1 + p T^{2}$$
3 $$1$$
good5$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
7$C_2^2$ $$1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
11$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
17$C_2^2$ $$1 + 10 T^{2} + p^{2} T^{4}$$
19$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
23$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
29$C_2^2$ $$1 + 46 T^{2} + p^{2} T^{4}$$
31$C_2^2$ $$1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
37$C_2$ $$( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} )$$
41$C_2^2$ $$1 + 58 T^{2} + p^{2} T^{4}$$
43$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
47$C_2^2$ $$1 + 46 T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 + 94 T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 + 22 T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
67$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
71$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
73$C_2^2$ $$1 - 2 T^{2} + p^{2} T^{4}$$
79$C_2^2$ $$1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4}$$
83$C_2^2$ $$1 - 50 T^{2} + p^{2} T^{4}$$
89$C_2^2$ $$1 + 82 T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 - p T^{2} )^{2}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Imaginary part of the first few zeros on the critical line

−15.6573552799, −15.0493121231, −14.7989083334, −14.1265649677, −13.9792602007, −13.3186411952, −12.8699612441, −12.5414157211, −11.7734084601, −11.3403320356, −10.8796379876, −10.2435328648, −9.72560491070, −9.21164953256, −8.64977247819, −8.15104498432, −7.75993010545, −6.99953147484, −6.14051787937, −5.66287191424, −4.84339700748, −4.40255307769, −3.66989511421, −2.65205799832, −1.21843548350, 1.21843548350, 2.65205799832, 3.66989511421, 4.40255307769, 4.84339700748, 5.66287191424, 6.14051787937, 6.99953147484, 7.75993010545, 8.15104498432, 8.64977247819, 9.21164953256, 9.72560491070, 10.2435328648, 10.8796379876, 11.3403320356, 11.7734084601, 12.5414157211, 12.8699612441, 13.3186411952, 13.9792602007, 14.1265649677, 14.7989083334, 15.0493121231, 15.6573552799