# Properties

 Label 8-1078e4-1.1-c1e4-0-8 Degree $8$ Conductor $1.350\times 10^{12}$ Sign $1$ Analytic cond. $5490.14$ Root an. cond. $2.93391$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s + 4-s − 2·8-s + 4·9-s + 2·11-s − 4·16-s + 8·18-s + 4·22-s + 4·23-s + 8·25-s − 24·29-s − 2·32-s + 4·36-s + 4·37-s + 16·43-s + 2·44-s + 8·46-s + 16·50-s + 24·53-s − 48·58-s + 3·64-s + 4·67-s − 40·71-s − 8·72-s + 8·74-s + 16·79-s + 9·81-s + ⋯
 L(s)  = 1 + 1.41·2-s + 1/2·4-s − 0.707·8-s + 4/3·9-s + 0.603·11-s − 16-s + 1.88·18-s + 0.852·22-s + 0.834·23-s + 8/5·25-s − 4.45·29-s − 0.353·32-s + 2/3·36-s + 0.657·37-s + 2.43·43-s + 0.301·44-s + 1.17·46-s + 2.26·50-s + 3.29·53-s − 6.30·58-s + 3/8·64-s + 0.488·67-s − 4.74·71-s − 0.942·72-s + 0.929·74-s + 1.80·79-s + 81-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 7^{8} \cdot 11^{4}$$ Sign: $1$ Analytic conductor: $$5490.14$$ Root analytic conductor: $$2.93391$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 7^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$7.099227553$$ $$L(\frac12)$$ $$\approx$$ $$7.099227553$$ $$L(\frac{3}{2})$$ 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)$
bad2$C_2$ $$( 1 - T + T^{2} )^{2}$$
7 $$1$$
11$C_2$ $$( 1 - T + T^{2} )^{2}$$
good3$C_2^3$ $$1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}$$
5$C_2^3$ $$1 - 8 T^{2} + 39 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8}$$
13$C_2^2$ $$( 1 + 18 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^3$ $$1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2^3$ $$1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8}$$
23$C_2^2$ $$( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
31$C_2^3$ $$1 - 60 T^{2} + 2639 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8}$$
37$C_2^2$ $$( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 50 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 - 4 T + p T^{2} )^{4}$$
47$C_2^3$ $$1 - 76 T^{2} + 3567 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
59$C_2^3$ $$1 - 100 T^{2} + 6519 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^3$ $$1 - 90 T^{2} + 4379 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8}$$
67$C_2^2$ $$( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2$ $$( 1 + 10 T + p T^{2} )^{4}$$
73$C_2^3$ $$1 + 54 T^{2} - 2413 T^{4} + 54 p^{2} T^{6} + p^{4} T^{8}$$
79$C_2^2$ $$( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 + 158 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^3$ $$1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2^2$ $$( 1 - 48 T^{2} + p^{2} T^{4} )^{2}$$
show less
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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

−6.98599038472824670738269954011, −6.87814900858078444822581643881, −6.60033486625472391882394591366, −6.41612098810812933989532338711, −5.99810773635683109634343046609, −5.86302115455856576529990933302, −5.59123617697526498404469089651, −5.54364405496541873922930200555, −5.26135419763845999217246132757, −5.06790717818188946222630863935, −4.70370248902019579022030748297, −4.35159464175674344870093384405, −4.22470240037869523938341830626, −4.21907368570579354291815534310, −3.78314046931653629738575030484, −3.66286346539957141566501866204, −3.47508202460065744691953011093, −2.90454780051418567248627330187, −2.82963158174833831468337548854, −2.47782074971000954998203089855, −2.08252570246507862304766255653, −1.73414405094382841272524904422, −1.43254125869397259724395104169, −0.939337805131261358351192474322, −0.50556243447685142525612914078, 0.50556243447685142525612914078, 0.939337805131261358351192474322, 1.43254125869397259724395104169, 1.73414405094382841272524904422, 2.08252570246507862304766255653, 2.47782074971000954998203089855, 2.82963158174833831468337548854, 2.90454780051418567248627330187, 3.47508202460065744691953011093, 3.66286346539957141566501866204, 3.78314046931653629738575030484, 4.21907368570579354291815534310, 4.22470240037869523938341830626, 4.35159464175674344870093384405, 4.70370248902019579022030748297, 5.06790717818188946222630863935, 5.26135419763845999217246132757, 5.54364405496541873922930200555, 5.59123617697526498404469089651, 5.86302115455856576529990933302, 5.99810773635683109634343046609, 6.41612098810812933989532338711, 6.60033486625472391882394591366, 6.87814900858078444822581643881, 6.98599038472824670738269954011