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Properties

Label 8-2541e4-1.1-c1e4-0-0
Degree $8$
Conductor $4.169\times 10^{13}$
Sign $1$
Analytic cond. $169483.$
Root an. cond. $4.50444$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·3-s + 6·5-s − 8·6-s + 4·7-s − 4·8-s + 10·9-s + 12·10-s − 2·13-s + 8·14-s − 24·15-s − 6·16-s + 2·17-s + 20·18-s + 6·19-s − 16·21-s + 10·23-s + 16·24-s + 8·25-s − 4·26-s − 20·27-s + 14·29-s − 48·30-s + 4·34-s + 24·35-s − 12·37-s + 12·38-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.30·3-s + 2.68·5-s − 3.26·6-s + 1.51·7-s − 1.41·8-s + 10/3·9-s + 3.79·10-s − 0.554·13-s + 2.13·14-s − 6.19·15-s − 3/2·16-s + 0.485·17-s + 4.71·18-s + 1.37·19-s − 3.49·21-s + 2.08·23-s + 3.26·24-s + 8/5·25-s − 0.784·26-s − 3.84·27-s + 2.59·29-s − 8.76·30-s + 0.685·34-s + 4.05·35-s − 1.97·37-s + 1.94·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(169483.\)
Root analytic conductor: \(4.50444\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(11.41819365\)
\(L(\frac12)\) \(\approx\) \(11.41819365\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{4} \)
7$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + p^{2} T^{2} - p^{2} T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 28 T^{2} - 84 T^{3} + 219 T^{4} - 84 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 12 T^{2} + 46 T^{3} + 332 T^{4} + 46 p T^{5} + 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 56 T^{2} - 80 T^{3} + 1339 T^{4} - 80 p T^{5} + 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 72 T^{2} - 330 T^{3} + 2012 T^{4} - 330 p T^{5} + 72 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 116 T^{2} - 694 T^{3} + 4276 T^{4} - 694 p T^{5} + 116 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 14 T + 164 T^{2} - 1238 T^{3} + 7756 T^{4} - 1238 p T^{5} + 164 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 108 T^{2} - 24 T^{3} + 4766 T^{4} - 24 p T^{5} + 108 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 168 T^{2} + 1284 T^{3} + 9650 T^{4} + 1284 p T^{5} + 168 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 196 T^{2} - 1712 T^{3} + 11994 T^{4} - 1712 p T^{5} + 196 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 306 T^{2} - 2920 T^{3} + 22895 T^{4} - 2920 p T^{5} + 306 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 88 T^{2} - 568 T^{3} + 4023 T^{4} - 568 p T^{5} + 88 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 244 T^{2} + 2096 T^{3} + 18582 T^{4} + 2096 p T^{5} + 244 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 2 T - 20 T^{2} - 136 T^{3} + 105 p T^{4} - 136 p T^{5} - 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 180 T^{2} - 418 T^{3} + 14804 T^{4} - 418 p T^{5} + 180 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 282 T^{2} + 2776 T^{3} + 24707 T^{4} + 2776 p T^{5} + 282 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 76 T^{2} - 790 T^{3} - 8172 T^{4} - 790 p T^{5} + 76 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 52 T^{2} + 406 T^{3} + 7828 T^{4} + 406 p T^{5} + 52 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 312 T^{2} - 3272 T^{3} + 37490 T^{4} - 3272 p T^{5} + 312 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 400 T^{2} - 4404 T^{3} + 52635 T^{4} - 4404 p T^{5} + 400 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 14 T + 244 T^{2} + 2764 T^{3} + 32067 T^{4} + 2764 p T^{5} + 244 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 38 T + 844 T^{2} - 12458 T^{3} + 140980 T^{4} - 12458 p T^{5} + 844 p^{2} T^{6} - 38 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.14389306724593840532814046303, −6.00064150197103967236295116082, −5.76086588237935038692059704559, −5.71417647352237932380863379773, −5.63774293758151513696039442888, −5.04692205160898939093093071304, −5.03580297697468732193820032783, −5.01475565579620461789687548907, −4.94678474437504982025975567405, −4.52257664539468126607537400659, −4.31664005739085178402835700868, −4.26705997544784764621437540156, −4.22137552032149407274242519403, −3.41438095539538484658937066538, −3.35357095779426986087208774080, −3.13070753791645009548920815348, −2.85287433775151158513803708994, −2.43810571941365077208032992883, −2.24145331418791110731341593944, −1.97712018464318716506515110123, −1.68605987517153223799192284958, −1.40916677278520374297909171318, −0.989476389642622289067440430397, −0.792600671453154317419519429089, −0.57794194531910888766695245795, 0.57794194531910888766695245795, 0.792600671453154317419519429089, 0.989476389642622289067440430397, 1.40916677278520374297909171318, 1.68605987517153223799192284958, 1.97712018464318716506515110123, 2.24145331418791110731341593944, 2.43810571941365077208032992883, 2.85287433775151158513803708994, 3.13070753791645009548920815348, 3.35357095779426986087208774080, 3.41438095539538484658937066538, 4.22137552032149407274242519403, 4.26705997544784764621437540156, 4.31664005739085178402835700868, 4.52257664539468126607537400659, 4.94678474437504982025975567405, 5.01475565579620461789687548907, 5.03580297697468732193820032783, 5.04692205160898939093093071304, 5.63774293758151513696039442888, 5.71417647352237932380863379773, 5.76086588237935038692059704559, 6.00064150197103967236295116082, 6.14389306724593840532814046303

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