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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

Origins of factors

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Normalization:  

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

Graph of the $Z$-function along the critical line