↑

Properties

Label 8-105e4-1.1-c3e4-0-2
Degree $8$
Conductor $121550625$
Sign $1$
Analytic cond. $1473.06$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 6·3-s + 18·4-s + 10·5-s + 24·6-s + 50·7-s + 88·8-s + 9·9-s + 40·10-s + 32·11-s + 108·12-s + 28·13-s + 200·14-s + 60·15-s + 288·16-s + 20·17-s + 36·18-s − 18·19-s + 180·20-s + 300·21-s + 128·22-s + 68·23-s + 528·24-s + 25·25-s + 112·26-s − 54·27-s + 900·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 9/4·4-s + 0.894·5-s + 1.63·6-s + 2.69·7-s + 3.88·8-s + 1/3·9-s + 1.26·10-s + 0.877·11-s + 2.59·12-s + 0.597·13-s + 3.81·14-s + 1.03·15-s + 9/2·16-s + 0.285·17-s + 0.471·18-s − 0.217·19-s + 2.01·20-s + 3.11·21-s + 1.24·22-s + 0.616·23-s + 4.49·24-s + 1/5·25-s + 0.844·26-s − 0.384·27-s + 6.07·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1473.06\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(34.26290088\)
\(L(\frac12)\) \(\approx\) \(34.26290088\)
\(L(\frac{5}{2})\) 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$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
5$C_2$ \( ( 1 - p T + p^{2} T^{2} )^{2} \)
7$C_2^2$ \( 1 - 50 T + 177 p T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \)
good2$D_4\times C_2$ \( 1 - p^{2} T - p T^{2} - p^{3} T^{3} + 33 p^{2} T^{4} - p^{6} T^{5} - p^{7} T^{6} - p^{11} T^{7} + p^{12} T^{8} \)
11$D_4\times C_2$ \( 1 - 32 T + 994 T^{2} + 84224 T^{3} - 3133605 T^{4} + 84224 p^{3} T^{5} + 994 p^{6} T^{6} - 32 p^{9} T^{7} + p^{12} T^{8} \)
13$D_{4}$ \( ( 1 - 14 T + 3091 T^{2} - 14 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 20 T - 7214 T^{2} + 44240 T^{3} + 31870227 T^{4} + 44240 p^{3} T^{5} - 7214 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \)
19$D_4\times C_2$ \( 1 + 18 T - 5283 T^{2} - 145998 T^{3} - 17829748 T^{4} - 145998 p^{3} T^{5} - 5283 p^{6} T^{6} + 18 p^{9} T^{7} + p^{12} T^{8} \)
23$D_4\times C_2$ \( 1 - 68 T - 19514 T^{2} + 13328 T^{3} + 378346947 T^{4} + 13328 p^{3} T^{5} - 19514 p^{6} T^{6} - 68 p^{9} T^{7} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 332 T + 76262 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_4\times C_2$ \( 1 + 66 T - 1733 p T^{2} - 99198 T^{3} + 2355896964 T^{4} - 99198 p^{3} T^{5} - 1733 p^{7} T^{6} + 66 p^{9} T^{7} + p^{12} T^{8} \)
37$D_4\times C_2$ \( 1 + 18 T - 2495 T^{2} - 1772766 T^{3} - 2574191220 T^{4} - 1772766 p^{3} T^{5} - 2495 p^{6} T^{6} + 18 p^{9} T^{7} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 - 152 T + 113850 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_{4}$ \( ( 1 + 842 T + 333367 T^{2} + 842 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 212 T - 145138 T^{2} + 3723568 T^{3} + 20685730483 T^{4} + 3723568 p^{3} T^{5} - 145138 p^{6} T^{6} - 212 p^{9} T^{7} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 368 T - 189914 T^{2} - 10150912 T^{3} + 63518934267 T^{4} - 10150912 p^{3} T^{5} - 189914 p^{6} T^{6} - 368 p^{9} T^{7} + p^{12} T^{8} \)
59$D_4\times C_2$ \( 1 - 140 T - 326866 T^{2} + 9000880 T^{3} + 73832727115 T^{4} + 9000880 p^{3} T^{5} - 326866 p^{6} T^{6} - 140 p^{9} T^{7} + p^{12} T^{8} \)
61$D_4\times C_2$ \( 1 - 12 p T - 44894 T^{2} - 1521072 p T^{3} + 164091153819 T^{4} - 1521072 p^{4} T^{5} - 44894 p^{6} T^{6} - 12 p^{10} T^{7} + p^{12} T^{8} \)
67$D_4\times C_2$ \( 1 + 1066 T + 375741 T^{2} + 169588874 T^{3} + 146302956428 T^{4} + 169588874 p^{3} T^{5} + 375741 p^{6} T^{6} + 1066 p^{9} T^{7} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 + 1208 T + 959606 T^{2} + 1208 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 + 1654 T + 1273953 T^{2} + 1130887766 T^{3} + 921176810708 T^{4} + 1130887766 p^{3} T^{5} + 1273953 p^{6} T^{6} + 1654 p^{9} T^{7} + p^{12} T^{8} \)
79$D_4\times C_2$ \( 1 + 1134 T - 19563 T^{2} + 362246094 T^{3} + 827265943556 T^{4} + 362246094 p^{3} T^{5} - 19563 p^{6} T^{6} + 1134 p^{9} T^{7} + p^{12} T^{8} \)
83$D_{4}$ \( ( 1 - 968 T + 1357022 T^{2} - 968 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
89$D_4\times C_2$ \( 1 + 204 T + 809506 T^{2} - 444276912 T^{3} + 63335771235 T^{4} - 444276912 p^{3} T^{5} + 809506 p^{6} T^{6} + 204 p^{9} T^{7} + p^{12} T^{8} \)
97$D_{4}$ \( ( 1 - 1692 T + 1457670 T^{2} - 1692 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
show more
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

−9.890352268121026003523602124212, −9.159070891880381718184038088972, −8.836571760796948562540594622901, −8.816941666724778814027593661030, −8.633073251040782659702704994703, −7.942505448421718532537440712873, −7.75670088136346281834781742227, −7.52760226999496457116757489423, −7.46690107907103443585912685543, −6.93154525772611353904108982567, −6.76652282662650140758804102738, −6.21652168357746155122713979791, −5.71708238182665714472825365918, −5.48684992396402313529398169899, −5.40756857740769675729638464007, −4.77534861555684313803703888619, −4.56080314002054701644177843834, −4.17163824679879880479158155209, −3.83055352947834909581984429061, −3.26434154893318483568562906786, −3.02852779165894346235817469897, −1.95939343323169667836582251398, −1.87841712334366198302202168712, −1.73164909618390737586093779820, −1.42165646441822733232449095721, 1.42165646441822733232449095721, 1.73164909618390737586093779820, 1.87841712334366198302202168712, 1.95939343323169667836582251398, 3.02852779165894346235817469897, 3.26434154893318483568562906786, 3.83055352947834909581984429061, 4.17163824679879880479158155209, 4.56080314002054701644177843834, 4.77534861555684313803703888619, 5.40756857740769675729638464007, 5.48684992396402313529398169899, 5.71708238182665714472825365918, 6.21652168357746155122713979791, 6.76652282662650140758804102738, 6.93154525772611353904108982567, 7.46690107907103443585912685543, 7.52760226999496457116757489423, 7.75670088136346281834781742227, 7.942505448421718532537440712873, 8.633073251040782659702704994703, 8.816941666724778814027593661030, 8.836571760796948562540594622901, 9.159070891880381718184038088972, 9.890352268121026003523602124212

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