# Properties

 Label 8-189e4-1.1-c1e4-0-3 Degree $8$ Conductor $1275989841$ Sign $1$ Analytic cond. $5.18747$ Root an. cond. $1.22848$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·4-s + 10·7-s + 4·16-s + 6·19-s + 4·25-s − 20·28-s − 18·31-s + 8·37-s − 28·43-s + 61·49-s − 18·61-s − 16·64-s − 4·67-s + 42·73-s − 12·76-s + 8·79-s − 8·100-s + 48·103-s − 34·109-s + 40·112-s + 10·121-s + 36·124-s + 127-s + 131-s + 60·133-s + 137-s + 139-s + ⋯
 L(s)  = 1 − 4-s + 3.77·7-s + 16-s + 1.37·19-s + 4/5·25-s − 3.77·28-s − 3.23·31-s + 1.31·37-s − 4.26·43-s + 61/7·49-s − 2.30·61-s − 2·64-s − 0.488·67-s + 4.91·73-s − 1.37·76-s + 0.900·79-s − 4/5·100-s + 4.72·103-s − 3.25·109-s + 3.77·112-s + 0.909·121-s + 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 5.20·133-s + 0.0854·137-s + 0.0848·139-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$3^{12} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$5.18747$$ Root analytic conductor: $$1.22848$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{12} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.962349578$$ $$L(\frac12)$$ $$\approx$$ $$1.962349578$$ $$L(\frac{3}{2})$$ 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 $$1$$
7$C_2$ $$( 1 - 5 T + p T^{2} )^{2}$$
good2$C_2$$\times$$C_2^2$ $$( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} )$$
5$C_2^3$ $$1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^3$ $$1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8}$$
13$C_2^2$ $$( 1 - 14 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^3$ $$1 - 28 T^{2} + 495 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2^2$ $$( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
23$C_2^3$ $$1 - 4 T^{2} - 513 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 - 56 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 76 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 + 7 T + p T^{2} )^{4}$$
47$C_2^3$ $$1 - 40 T^{2} - 609 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^3$ $$1 + 104 T^{2} + 8007 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^3$ $$1 + 98 T^{2} + 6123 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 - 134 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^2$ $$( 1 - 21 T + 220 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2}$$
79$C_2$ $$( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2}$$
83$C_2^2$ $$( 1 + 70 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^3$ $$1 - 172 T^{2} + 21663 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2^2$ $$( 1 - 119 T^{2} + p^{2} T^{4} )^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−9.196388511497976356199062169067, −8.876555614053637804120198106225, −8.519379868362428858343814575190, −8.463681845534553039326776225981, −8.238439029753145795952760410200, −7.68868794836723936737761501634, −7.65998398712494571806786748335, −7.59746714390585447706212114441, −7.33341581612845252273168719148, −6.70539426307136703355588343480, −6.52859308162620839699364821736, −5.83485878016172540859065191283, −5.80396961269920167739436713063, −5.11314226612979465325326970094, −5.05064475941037451959452346539, −4.99544349149552491414040248634, −4.88792383534151970113164416805, −4.28653418143323724872235954554, −3.93012448432945952561673779543, −3.43660598060250442539838764466, −3.36655132950682273292083537907, −2.43187754419772588526089620704, −1.90748579708039254720588099870, −1.58810241130337706455791704347, −1.11851154836978925579362295626, 1.11851154836978925579362295626, 1.58810241130337706455791704347, 1.90748579708039254720588099870, 2.43187754419772588526089620704, 3.36655132950682273292083537907, 3.43660598060250442539838764466, 3.93012448432945952561673779543, 4.28653418143323724872235954554, 4.88792383534151970113164416805, 4.99544349149552491414040248634, 5.05064475941037451959452346539, 5.11314226612979465325326970094, 5.80396961269920167739436713063, 5.83485878016172540859065191283, 6.52859308162620839699364821736, 6.70539426307136703355588343480, 7.33341581612845252273168719148, 7.59746714390585447706212114441, 7.65998398712494571806786748335, 7.68868794836723936737761501634, 8.238439029753145795952760410200, 8.463681845534553039326776225981, 8.519379868362428858343814575190, 8.876555614053637804120198106225, 9.196388511497976356199062169067