# Properties

 Label 4-855e2-1.1-c1e2-0-6 Degree $4$ Conductor $731025$ Sign $-1$ Analytic cond. $46.6107$ Root an. cond. $2.61289$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 − 3·4-s + 5·16-s − 4·19-s − 25-s + 8·43-s − 14·49-s + 4·61-s − 3·64-s + 20·73-s + 12·76-s + 3·100-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s − 24·172-s + 173-s + 179-s + 181-s + 191-s + ⋯
 L(s)  = 1 − 3/2·4-s + 5/4·16-s − 0.917·19-s − 1/5·25-s + 1.21·43-s − 2·49-s + 0.512·61-s − 3/8·64-s + 2.34·73-s + 1.37·76-s + 3/10·100-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s − 1.82·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$731025$$    =    $$3^{4} \cdot 5^{2} \cdot 19^{2}$$ Sign: $-1$ Analytic conductor: $$46.6107$$ Root analytic conductor: $$2.61289$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(4,\ 731025,\ (\ :1/2, 1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad3 $$1$$
5$C_2$ $$1 + T^{2}$$
19$C_2$ $$1 + 4 T + p T^{2}$$
good2$C_2$ $$( 1 - T + p T^{2} )( 1 + T + p T^{2} )$$
7$C_2$ $$( 1 + p T^{2} )^{2}$$
11$C_2^2$ $$1 - 6 T^{2} + p^{2} T^{4}$$
13$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
17$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
23$C_2$ $$( 1 - p T^{2} )^{2}$$
29$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
41$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
47$C_2^2$ $$1 - 30 T^{2} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
59$C_2$ $$( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
61$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
67$C_2$ $$( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
71$C_2$ $$( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )$$
73$C_2$ $$( 1 - 10 T + p T^{2} )^{2}$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2^2$ $$1 - 22 T^{2} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )$$
97$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} )$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−8.277973675841816239455228595843, −7.74415933620971995578395538835, −7.30092387852484149612701981622, −6.59771949886092311755977163671, −6.31772517182021644904628485653, −5.69423999100043150300968189227, −5.24361911723200874500658764166, −4.72856937868326294980636111340, −4.43241097382010054660439073995, −3.79236167186905386391550194299, −3.48722200346027363540172637161, −2.62640101086036899692230571679, −1.94452644213633590102628391910, −0.959422674505439198259115714954, 0, 0.959422674505439198259115714954, 1.94452644213633590102628391910, 2.62640101086036899692230571679, 3.48722200346027363540172637161, 3.79236167186905386391550194299, 4.43241097382010054660439073995, 4.72856937868326294980636111340, 5.24361911723200874500658764166, 5.69423999100043150300968189227, 6.31772517182021644904628485653, 6.59771949886092311755977163671, 7.30092387852484149612701981622, 7.74415933620971995578395538835, 8.277973675841816239455228595843