↑

Properties

Label 8-2645e4-1.1-c1e4-0-2
Degree $8$
Conductor $4.894\times 10^{13}$
Sign $1$
Analytic cond. $198980.$
Root an. cond. $4.59569$
Motivic weight $1$
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  − 2-s + 3-s − 3·4-s − 4·5-s − 6-s + 3·7-s + 2·8-s − 9-s + 4·10-s + 5·11-s − 3·12-s − 3·14-s − 4·15-s + 5·16-s − 5·17-s + 18-s + 4·19-s + 12·20-s + 3·21-s − 5·22-s + 2·24-s + 10·25-s − 7·27-s − 9·28-s − 5·29-s + 4·30-s − 13·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 3/2·4-s − 1.78·5-s − 0.408·6-s + 1.13·7-s + 0.707·8-s − 1/3·9-s + 1.26·10-s + 1.50·11-s − 0.866·12-s − 0.801·14-s − 1.03·15-s + 5/4·16-s − 1.21·17-s + 0.235·18-s + 0.917·19-s + 2.68·20-s + 0.654·21-s − 1.06·22-s + 0.408·24-s + 2·25-s − 1.34·27-s − 1.70·28-s − 0.928·29-s + 0.730·30-s − 2.33·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(5^{4} \cdot 23^{8}\)
Sign: $1$
Analytic conductor: \(198980.\)
Root analytic conductor: \(4.59569\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 5^{4} \cdot 23^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.466666315\)
\(L(\frac12)\) \(\approx\) \(1.466666315\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
23 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + T + p^{2} T^{2} + 5 T^{3} + 5 p T^{4} + 5 p T^{5} + p^{4} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
3$C_2 \wr S_4$ \( 1 - T + 2 T^{2} + 4 T^{3} - 8 T^{4} + 4 p T^{5} + 2 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 3 T + 8 T^{2} - 26 T^{3} + 120 T^{4} - 26 p T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 5 T + 49 T^{2} - 163 T^{3} + 835 T^{4} - 163 p T^{5} + 49 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 23 T^{2} + 75 T^{3} + 214 T^{4} + 75 p T^{5} + 23 p^{2} T^{6} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 5 T + 32 T^{2} + 90 T^{3} + 584 T^{4} + 90 p T^{5} + 32 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 4 T + 27 T^{2} - 137 T^{3} + 357 T^{4} - 137 p T^{5} + 27 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 5 T + 53 T^{2} + 262 T^{3} + 2590 T^{4} + 262 p T^{5} + 53 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 13 T + 161 T^{2} + 1163 T^{3} + 7861 T^{4} + 1163 p T^{5} + 161 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 3 T + 110 T^{2} - 238 T^{3} + 5520 T^{4} - 238 p T^{5} + 110 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 20 T + 7 p T^{2} + 2739 T^{3} + 20345 T^{4} + 2739 p T^{5} + 7 p^{3} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 12 T + 205 T^{2} - 1529 T^{3} + 13844 T^{4} - 1529 p T^{5} + 205 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 9 T + 111 T^{2} + 524 T^{3} + 5012 T^{4} + 524 p T^{5} + 111 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 5 T + 118 T^{2} - 694 T^{3} + 7486 T^{4} - 694 p T^{5} + 118 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 4 T + 171 T^{2} + 791 T^{3} + 13404 T^{4} + 791 p T^{5} + 171 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 12 T + 278 T^{2} - 2176 T^{3} + 26415 T^{4} - 2176 p T^{5} + 278 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 18 T + 311 T^{2} - 2953 T^{3} + 444 p T^{4} - 2953 p T^{5} + 311 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 30 T + 585 T^{2} - 7567 T^{3} + 74555 T^{4} - 7567 p T^{5} + 585 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - T + 134 T^{2} - 420 T^{3} + 8898 T^{4} - 420 p T^{5} + 134 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 16 T + 278 T^{2} - 2984 T^{3} + 31759 T^{4} - 2984 p T^{5} + 278 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 24 T + 449 T^{2} - 5977 T^{3} + 60332 T^{4} - 5977 p T^{5} + 449 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 9 T + 266 T^{2} + 2012 T^{3} + 32324 T^{4} + 2012 p T^{5} + 266 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 39 T + 898 T^{2} - 13972 T^{3} + 159698 T^{4} - 13972 p T^{5} + 898 p^{2} T^{6} - 39 p^{3} T^{7} + p^{4} T^{8} \)
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

−6.34616278297364520548400329515, −6.29133133564270403922513713101, −5.73655281298159740989425024071, −5.55234313117020899901075840107, −5.52924714172021214769537279056, −4.98023660846433527250446082585, −4.93187436045095670541555860582, −4.73485532297086125442645873838, −4.69763313958471066914196567939, −4.64389372573703805109997509390, −3.87325327183494563179060774970, −3.75783665953335186122025844260, −3.69884246737429648309540017853, −3.68839480475978506561591202942, −3.56914501860388088665009558546, −3.31988566310060481304329193172, −2.81038240484537664323884626794, −2.39372643769629127049424505388, −2.06282063864684883097693012680, −2.00269063420354185314041174531, −1.80309113705861064954506103628, −1.26298819591276850086889357540, −0.69006237040525117425658863436, −0.57263983365369974681507648519, −0.44813050212737365218498837983, 0.44813050212737365218498837983, 0.57263983365369974681507648519, 0.69006237040525117425658863436, 1.26298819591276850086889357540, 1.80309113705861064954506103628, 2.00269063420354185314041174531, 2.06282063864684883097693012680, 2.39372643769629127049424505388, 2.81038240484537664323884626794, 3.31988566310060481304329193172, 3.56914501860388088665009558546, 3.68839480475978506561591202942, 3.69884246737429648309540017853, 3.75783665953335186122025844260, 3.87325327183494563179060774970, 4.64389372573703805109997509390, 4.69763313958471066914196567939, 4.73485532297086125442645873838, 4.93187436045095670541555860582, 4.98023660846433527250446082585, 5.52924714172021214769537279056, 5.55234313117020899901075840107, 5.73655281298159740989425024071, 6.29133133564270403922513713101, 6.34616278297364520548400329515

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