Properties

Label 8-429e4-1.1-c1e4-0-0
Degree $8$
Conductor $33871089681$
Sign $1$
Analytic cond. $137.701$
Root an. cond. $1.85083$
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  − 2·2-s − 4·3-s + 2·4-s + 8·6-s + 2·7-s + 10·9-s − 4·11-s − 8·12-s + 4·13-s − 4·14-s − 3·16-s − 8·17-s − 20·18-s + 18·19-s − 8·21-s + 8·22-s − 8·25-s − 8·26-s − 20·27-s + 4·28-s − 10·29-s + 12·31-s + 4·32-s + 16·33-s + 16·34-s + 20·36-s − 2·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s + 4-s + 3.26·6-s + 0.755·7-s + 10/3·9-s − 1.20·11-s − 2.30·12-s + 1.10·13-s − 1.06·14-s − 3/4·16-s − 1.94·17-s − 4.71·18-s + 4.12·19-s − 1.74·21-s + 1.70·22-s − 8/5·25-s − 1.56·26-s − 3.84·27-s + 0.755·28-s − 1.85·29-s + 2.15·31-s + 0.707·32-s + 2.78·33-s + 2.74·34-s + 10/3·36-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 11^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(137.701\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 11^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.4588779429\)
\(L(\frac12)\) \(\approx\) \(0.4588779429\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{4} \)
11$C_1$ \( ( 1 + T )^{4} \)
13$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr S_4$ \( 1 + p T + p T^{2} - T^{4} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 8 T^{2} - 14 T^{3} + 26 T^{4} - 14 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \)
7$C_2 \wr S_4$ \( 1 - 2 T + 12 T^{2} + 2 T^{3} + 54 T^{4} + 2 p T^{5} + 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 8 T + 42 T^{2} + 246 T^{3} + 1262 T^{4} + 246 p T^{5} + 42 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 18 T + 180 T^{2} - 1206 T^{3} + 6054 T^{4} - 1206 p T^{5} + 180 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 48 T^{2} + 148 T^{3} + 1022 T^{4} + 148 p T^{5} + 48 p^{2} T^{6} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 10 T + 86 T^{2} + 488 T^{3} + 110 p T^{4} + 488 p T^{5} + 86 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 12 T + 104 T^{2} - 658 T^{3} + 3558 T^{4} - 658 p T^{5} + 104 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 2 T + 108 T^{2} + 286 T^{3} + 5286 T^{4} + 286 p T^{5} + 108 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 2 T + 112 T^{2} - 106 T^{3} + 5734 T^{4} - 106 p T^{5} + 112 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 28 T + 374 T^{2} - 3342 T^{3} + 23802 T^{4} - 3342 p T^{5} + 374 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 6 T + 104 T^{2} - 982 T^{3} + 5486 T^{4} - 982 p T^{5} + 104 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 4 T + 132 T^{2} + 412 T^{3} + 8390 T^{4} + 412 p T^{5} + 132 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 16 T + 140 T^{2} - 128 T^{3} - 1898 T^{4} - 128 p T^{5} + 140 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 10 T + 164 T^{2} + 742 T^{3} + 9750 T^{4} + 742 p T^{5} + 164 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 180 T^{2} + 18 T^{3} + 16790 T^{4} + 18 p T^{5} + 180 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 10 T + 208 T^{2} - 1994 T^{3} + 19582 T^{4} - 1994 p T^{5} + 208 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 6 T + 184 T^{2} + 22 p T^{3} + 16118 T^{4} + 22 p^{2} T^{5} + 184 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 8 T + 66 T^{2} + 514 T^{3} + 1962 T^{4} + 514 p T^{5} + 66 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 8 T + 212 T^{2} + 1368 T^{3} + 24806 T^{4} + 1368 p T^{5} + 212 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 6 T + 348 T^{2} - 1544 T^{3} + 46058 T^{4} - 1544 p T^{5} + 348 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 10 T + 140 T^{2} + 802 T^{3} - 4490 T^{4} + 802 p T^{5} + 140 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
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

−7.86647572942243790026980120070, −7.68476205059694681298338172017, −7.57652671079220205408341653824, −7.45725741103890334986062156787, −7.34191823636082089700349327628, −6.89259130390625756110450785177, −6.62263700049102057003896726119, −6.31900800511209078782859840678, −6.01125711637533476655104168436, −5.83913809868824680606584748797, −5.47352530890130098351063332751, −5.46530010436867002387178544749, −5.27368577745143377462680955091, −4.71706306782640570491021449176, −4.49469340812319114559957848267, −4.35808684881984360956449430405, −4.03715281022682656299703928721, −3.51190101980306841362668213634, −3.22439467580610903491995591848, −2.65647465431278195067493538327, −2.22819009930689211436517355110, −1.87387695558354055611315603474, −1.27714062245721150583364961300, −0.974181413234358083181963502538, −0.50682446882024202201241971732, 0.50682446882024202201241971732, 0.974181413234358083181963502538, 1.27714062245721150583364961300, 1.87387695558354055611315603474, 2.22819009930689211436517355110, 2.65647465431278195067493538327, 3.22439467580610903491995591848, 3.51190101980306841362668213634, 4.03715281022682656299703928721, 4.35808684881984360956449430405, 4.49469340812319114559957848267, 4.71706306782640570491021449176, 5.27368577745143377462680955091, 5.46530010436867002387178544749, 5.47352530890130098351063332751, 5.83913809868824680606584748797, 6.01125711637533476655104168436, 6.31900800511209078782859840678, 6.62263700049102057003896726119, 6.89259130390625756110450785177, 7.34191823636082089700349327628, 7.45725741103890334986062156787, 7.57652671079220205408341653824, 7.68476205059694681298338172017, 7.86647572942243790026980120070

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