# Properties

 Label 8-1805e4-1.1-c1e4-0-0 Degree $8$ Conductor $1.061\times 10^{13}$ Sign $1$ Analytic cond. $43153.6$ Root an. cond. $3.79644$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2-s + 3·3-s − 4-s + 4·5-s + 3·6-s − 4·7-s + 2·8-s − 9-s + 4·10-s − 2·11-s − 3·12-s + 7·13-s − 4·14-s + 12·15-s + 3·16-s − 17-s − 18-s − 4·20-s − 12·21-s − 2·22-s + 2·23-s + 6·24-s + 10·25-s + 7·26-s − 11·27-s + 4·28-s − 29-s + ⋯
 L(s)  = 1 + 0.707·2-s + 1.73·3-s − 1/2·4-s + 1.78·5-s + 1.22·6-s − 1.51·7-s + 0.707·8-s − 1/3·9-s + 1.26·10-s − 0.603·11-s − 0.866·12-s + 1.94·13-s − 1.06·14-s + 3.09·15-s + 3/4·16-s − 0.242·17-s − 0.235·18-s − 0.894·20-s − 2.61·21-s − 0.426·22-s + 0.417·23-s + 1.22·24-s + 2·25-s + 1.37·26-s − 2.11·27-s + 0.755·28-s − 0.185·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$5^{4} \cdot 19^{8}$$ Sign: $1$ Analytic conductor: $$43153.6$$ Root analytic conductor: $$3.79644$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 5^{4} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$6.111227181$$ $$L(\frac12)$$ $$\approx$$ $$6.111227181$$ $$L(\frac{3}{2})$$ 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)$
bad5$C_1$ $$( 1 - T )^{4}$$
19 $$1$$
good2$C_2 \wr S_4$ $$1 - T + p T^{2} - 5 T^{3} + 3 p T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
3$C_2 \wr S_4$ $$1 - p T + 10 T^{2} - 22 T^{3} + 44 T^{4} - 22 p T^{5} + 10 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8}$$
7$C_2 \wr S_4$ $$1 + 4 T + 27 T^{2} + 69 T^{3} + 272 T^{4} + 69 p T^{5} + 27 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr S_4$ $$1 + 2 T + 19 T^{2} + 47 T^{3} + 179 T^{4} + 47 p T^{5} + 19 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2 \wr S_4$ $$1 - 7 T + 59 T^{2} - 250 T^{3} + 1180 T^{4} - 250 p T^{5} + 59 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8}$$
17$C_2 \wr S_4$ $$1 + T + 38 T^{2} + 4 p T^{3} + 822 T^{4} + 4 p^{2} T^{5} + 38 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
23$C_2 \wr S_4$ $$1 - 2 T + 75 T^{2} - 145 T^{3} + 2398 T^{4} - 145 p T^{5} + 75 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr S_4$ $$1 + T + 53 T^{2} + 281 T^{3} + 1251 T^{4} + 281 p T^{5} + 53 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr S_4$ $$1 + 57 T^{2} - 5 T^{3} + 2675 T^{4} - 5 p T^{5} + 57 p^{2} T^{6} + p^{4} T^{8}$$
37$C_2 \wr S_4$ $$1 + 2 T + 117 T^{2} + 99 T^{3} + 5802 T^{4} + 99 p T^{5} + 117 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr S_4$ $$1 + 8 T + 77 T^{2} + 13 T^{3} + 714 T^{4} + 13 p T^{5} + 77 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr S_4$ $$1 - T + 74 T^{2} + 68 T^{3} + 3460 T^{4} + 68 p T^{5} + 74 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr S_4$ $$1 + 12 T + 133 T^{2} + 763 T^{3} + 5768 T^{4} + 763 p T^{5} + 133 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr S_4$ $$1 + 5 T + 126 T^{2} + 568 T^{3} + 7684 T^{4} + 568 p T^{5} + 126 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr S_4$ $$1 + 5 T + 111 T^{2} + 385 T^{3} + 8011 T^{4} + 385 p T^{5} + 111 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr S_4$ $$1 + 114 T^{2} + 88 T^{3} + 9515 T^{4} + 88 p T^{5} + 114 p^{2} T^{6} + p^{4} T^{8}$$
67$C_2 \wr S_4$ $$1 - 4 T + 232 T^{2} - 828 T^{3} + 22174 T^{4} - 828 p T^{5} + 232 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr S_4$ $$1 - 20 T + 375 T^{2} - 4165 T^{3} + 42925 T^{4} - 4165 p T^{5} + 375 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr S_4$ $$1 + 20 T + 387 T^{2} + 57 p T^{3} + 44118 T^{4} + 57 p^{2} T^{5} + 387 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr S_4$ $$1 - 17 T + 388 T^{2} - 4009 T^{3} + 48638 T^{4} - 4009 p T^{5} + 388 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr S_4$ $$1 - T + 270 T^{2} - 194 T^{3} + 31408 T^{4} - 194 p T^{5} + 270 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr S_4$ $$1 - 11 T + 266 T^{2} - 1549 T^{3} + 27690 T^{4} - 1549 p T^{5} + 266 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}$$
97$C_2 \wr S_4$ $$1 - T + 122 T^{2} - 1096 T^{3} + 12292 T^{4} - 1096 p T^{5} + 122 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
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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

−6.44464758554663210699469466173, −6.41652300670887493039467820260, −6.19038164273225385999761287711, −6.03087779086326366838430068012, −5.89412080158017210107251601799, −5.32040464985557127612995396670, −5.26796861619405731763144706433, −5.06582444282989457149768264812, −5.05780752010583385045590106771, −4.56684970062580303927838719419, −4.56584503445334664662144746159, −4.11481977160444074387543562783, −3.69029952371709733739284118189, −3.60886012544964603176212143468, −3.41092448632297249398630390656, −3.25572831647002704529776690325, −2.97709438206567097364884155445, −2.86754082881879396300246703030, −2.53645489756928128075821024329, −2.08425907450006956228066763648, −1.96274365843339118862116095577, −1.79595525762831313268118671784, −1.28786942053371100008424776250, −0.953279651893298734050298226696, −0.30631676752128210527202595982, 0.30631676752128210527202595982, 0.953279651893298734050298226696, 1.28786942053371100008424776250, 1.79595525762831313268118671784, 1.96274365843339118862116095577, 2.08425907450006956228066763648, 2.53645489756928128075821024329, 2.86754082881879396300246703030, 2.97709438206567097364884155445, 3.25572831647002704529776690325, 3.41092448632297249398630390656, 3.60886012544964603176212143468, 3.69029952371709733739284118189, 4.11481977160444074387543562783, 4.56584503445334664662144746159, 4.56684970062580303927838719419, 5.05780752010583385045590106771, 5.06582444282989457149768264812, 5.26796861619405731763144706433, 5.32040464985557127612995396670, 5.89412080158017210107251601799, 6.03087779086326366838430068012, 6.19038164273225385999761287711, 6.41652300670887493039467820260, 6.44464758554663210699469466173