Properties

Label 8-605e4-1.1-c1e4-0-1
Degree $8$
Conductor $133974300625$
Sign $1$
Analytic cond. $544.665$
Root an. cond. $2.19794$
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 + 2·4-s − 5-s + 3·6-s + 3·7-s + 3·9-s − 10-s + 6·12-s − 4·13-s + 3·14-s − 3·15-s + 3·18-s − 4·19-s − 2·20-s + 9·21-s − 32·23-s − 4·26-s + 6·28-s − 6·29-s − 3·30-s + 2·31-s − 11·32-s − 3·35-s + 6·36-s + 8·37-s − 4·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 4-s − 0.447·5-s + 1.22·6-s + 1.13·7-s + 9-s − 0.316·10-s + 1.73·12-s − 1.10·13-s + 0.801·14-s − 0.774·15-s + 0.707·18-s − 0.917·19-s − 0.447·20-s + 1.96·21-s − 6.67·23-s − 0.784·26-s + 1.13·28-s − 1.11·29-s − 0.547·30-s + 0.359·31-s − 1.94·32-s − 0.507·35-s + 36-s + 1.31·37-s − 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(544.665\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{605} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.118454483\)
\(L(\frac12)\) \(\approx\) \(2.118454483\)
\(L(\frac{3}{2})\) 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$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
11 \( 1 \)
good2$C_4\times C_2$ \( 1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
3$C_4\times C_2$ \( 1 - p T + 2 p T^{2} - p^{2} T^{3} + p^{2} T^{4} - p^{3} T^{5} + 2 p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
7$C_4\times C_2$ \( 1 - 3 T + 2 T^{2} + 15 T^{3} - 59 T^{4} + 15 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_4\times C_2$ \( 1 + 4 T + 3 T^{2} - 40 T^{3} - 199 T^{4} - 40 p T^{5} + 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_4\times C_2$ \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \)
19$C_4\times C_2$ \( 1 + 4 T - 3 T^{2} - 88 T^{3} - 295 T^{4} - 88 p T^{5} - 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{4} \)
29$C_4\times C_2$ \( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 132 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_4\times C_2$ \( 1 - 2 T - 27 T^{2} + 116 T^{3} + 605 T^{4} + 116 p T^{5} - 27 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
37$C_4\times C_2$ \( 1 - 8 T + 27 T^{2} + 80 T^{3} - 1639 T^{4} + 80 p T^{5} + 27 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
41$C_4\times C_2$ \( 1 - 5 T - 16 T^{2} + 285 T^{3} - 769 T^{4} + 285 p T^{5} - 16 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{4} \)
47$C_4\times C_2$ \( 1 - 3 T - 38 T^{2} + 255 T^{3} + 1021 T^{4} + 255 p T^{5} - 38 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
53$C_4\times C_2$ \( 1 + 4 T - 37 T^{2} - 360 T^{3} + 521 T^{4} - 360 p T^{5} - 37 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_4\times C_2$ \( 1 - 2 T - 55 T^{2} + 228 T^{3} + 2789 T^{4} + 228 p T^{5} - 55 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_4\times C_2$ \( 1 - 11 T + 60 T^{2} + 11 T^{3} - 3781 T^{4} + 11 p T^{5} + 60 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2$ \( ( 1 + 13 T + p T^{2} )^{4} \)
71$C_4\times C_2$ \( 1 + 2 T - 67 T^{2} - 276 T^{3} + 4205 T^{4} - 276 p T^{5} - 67 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_4\times C_2$ \( 1 - 8 T - 9 T^{2} + 656 T^{3} - 4591 T^{4} + 656 p T^{5} - 9 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
79$C_4\times C_2$ \( 1 + 10 T + 21 T^{2} - 580 T^{3} - 7459 T^{4} - 580 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_4\times C_2$ \( 1 + 4 T - 67 T^{2} - 600 T^{3} + 3161 T^{4} - 600 p T^{5} - 67 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2$ \( ( 1 - T + p T^{2} )^{4} \)
97$C_4\times C_2$ \( 1 - 8 T - 33 T^{2} + 1040 T^{3} - 5119 T^{4} + 1040 p T^{5} - 33 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.61880610988925005955761734586, −7.56909427982818580786916189681, −7.33949572748974404850200086492, −7.32442127702582878885075089311, −6.65398580025625962532903583065, −6.41085001414077972736575446037, −6.18534576004986685578290837291, −5.91131485224818000266982901215, −5.72032501444492930582601360601, −5.65317208821220037234345754557, −5.46670749696114166845279095066, −4.79965826237156350001113941975, −4.40238840880682856611799400786, −4.23339119669243075436047087455, −4.15487183417568456685304694854, −3.97231325903907202300397747043, −3.76351380540971861461186752261, −3.26935150372612190660382129586, −2.72756225843641035421928372655, −2.60996089031176532691576586150, −2.52442023018752714886391843180, −1.91095190029464593981894167460, −1.89075250441913589942215105599, −1.64543658180506032343725931973, −0.27669842401542515469935513239, 0.27669842401542515469935513239, 1.64543658180506032343725931973, 1.89075250441913589942215105599, 1.91095190029464593981894167460, 2.52442023018752714886391843180, 2.60996089031176532691576586150, 2.72756225843641035421928372655, 3.26935150372612190660382129586, 3.76351380540971861461186752261, 3.97231325903907202300397747043, 4.15487183417568456685304694854, 4.23339119669243075436047087455, 4.40238840880682856611799400786, 4.79965826237156350001113941975, 5.46670749696114166845279095066, 5.65317208821220037234345754557, 5.72032501444492930582601360601, 5.91131485224818000266982901215, 6.18534576004986685578290837291, 6.41085001414077972736575446037, 6.65398580025625962532903583065, 7.32442127702582878885075089311, 7.33949572748974404850200086492, 7.56909427982818580786916189681, 7.61880610988925005955761734586

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