# Properties

 Degree 4 Conductor $3^{2} \cdot 7^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − 3·3-s − 2·4-s + 7-s + 6·9-s + 6·12-s − 9·19-s − 3·21-s + 5·25-s − 9·27-s − 2·28-s + 15·31-s − 12·36-s − 37-s − 10·43-s − 6·49-s + 27·57-s + 12·61-s + 6·63-s + 8·64-s − 11·67-s − 27·73-s − 15·75-s + 18·76-s + 13·79-s + 9·81-s + 6·84-s − 45·93-s + ⋯
 L(s)  = 1 − 1.73·3-s − 4-s + 0.377·7-s + 2·9-s + 1.73·12-s − 2.06·19-s − 0.654·21-s + 25-s − 1.73·27-s − 0.377·28-s + 2.69·31-s − 2·36-s − 0.164·37-s − 1.52·43-s − 6/7·49-s + 3.57·57-s + 1.53·61-s + 0.755·63-s + 64-s − 1.34·67-s − 3.16·73-s − 1.73·75-s + 2.06·76-s + 1.46·79-s + 81-s + 0.654·84-s − 4.66·93-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$441$$    =    $$3^{2} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : $\chi_{441} (1, \cdot )$ Sato-Tate : $\mathrm{U}(1)$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 441,\ (\ :1/2, 1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$0.2392939029$$ $$L(\frac12)$$ $$\approx$$ $$0.2392939029$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;7\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ $$1 + p T + p T^{2}$$
7$C_2$ $$1 - T + p T^{2}$$
good2$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
5$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
11$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} )$$
17$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
19$C_2$ $$( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
29$C_2$ $$( 1 - p T^{2} )^{2}$$
31$C_2$ $$( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
37$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} )$$
41$C_2$ $$( 1 + p T^{2} )^{2}$$
43$C_2$ $$( 1 + 5 T + p T^{2} )^{2}$$
47$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
53$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
59$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
61$C_2$ $$( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} )$$
67$C_2$ $$( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
71$C_2$ $$( 1 - p T^{2} )^{2}$$
73$C_2$ $$( 1 + 10 T + p T^{2} )( 1 + 17 T + p T^{2} )$$
79$C_2$ $$( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
83$C_2$ $$( 1 + p T^{2} )^{2}$$
89$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
97$C_2$ $$( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} )$$
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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

−15.55152607378139954871197988100, −14.86571197707096909151568063946, −14.07800191195924878406018486295, −13.01262643877904661435148088220, −12.97159492547354371966127994182, −11.83685127878360589018873821263, −11.49613855591697476522057219719, −10.37061436856425034139712770950, −10.23518579966480607595970757355, −8.914387133758119867741938583574, −8.219434135148846900962796918633, −6.79345536931461537031444424174, −6.16523471638974682138318287764, −4.87678738065384911829484710659, −4.46412573016302803392609904678, 4.46412573016302803392609904678, 4.87678738065384911829484710659, 6.16523471638974682138318287764, 6.79345536931461537031444424174, 8.219434135148846900962796918633, 8.914387133758119867741938583574, 10.23518579966480607595970757355, 10.37061436856425034139712770950, 11.49613855591697476522057219719, 11.83685127878360589018873821263, 12.97159492547354371966127994182, 13.01262643877904661435148088220, 14.07800191195924878406018486295, 14.86571197707096909151568063946, 15.55152607378139954871197988100