Properties

Label 8-2541e4-1.1-c1e4-0-5
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 $4$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·3-s − 2·5-s − 8·6-s + 4·7-s + 4·8-s + 10·9-s + 4·10-s − 10·13-s − 8·14-s − 8·15-s − 6·16-s − 6·17-s − 20·18-s − 18·19-s + 16·21-s − 2·23-s + 16·24-s − 12·25-s + 20·26-s + 20·27-s − 6·29-s + 16·30-s + 12·34-s − 8·35-s − 4·37-s + 36·38-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.30·3-s − 0.894·5-s − 3.26·6-s + 1.51·7-s + 1.41·8-s + 10/3·9-s + 1.26·10-s − 2.77·13-s − 2.13·14-s − 2.06·15-s − 3/2·16-s − 1.45·17-s − 4.71·18-s − 4.12·19-s + 3.49·21-s − 0.417·23-s + 3.26·24-s − 2.39·25-s + 3.92·26-s + 3.84·27-s − 1.11·29-s + 2.92·30-s + 2.05·34-s − 1.35·35-s − 0.657·37-s + 5.83·38-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: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 16 T^{2} + 28 T^{3} + 111 T^{4} + 28 p T^{5} + 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 84 T^{2} + 422 T^{3} + 1844 T^{4} + 422 p T^{5} + 84 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 28 T^{2} + 24 T^{3} + 111 T^{4} + 24 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 184 T^{2} + 1278 T^{3} + 6468 T^{4} + 1278 p T^{5} + 184 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 28 T^{2} + 190 T^{3} + 516 T^{4} + 190 p T^{5} + 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 20 T^{2} + 126 T^{3} + 1404 T^{4} + 126 p T^{5} + 20 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 28 T^{2} - 216 T^{3} + 606 T^{4} - 216 p T^{5} + 28 p^{2} T^{6} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 72 T^{2} + 284 T^{3} + 4082 T^{4} + 284 p T^{5} + 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 124 T^{2} - 48 T^{3} + 6810 T^{4} - 48 p T^{5} + 124 p^{2} T^{6} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 282 T^{2} + 2632 T^{3} + 19943 T^{4} + 2632 p T^{5} + 282 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 44 T^{2} + 504 T^{3} + 93 p T^{4} + 504 p T^{5} + 44 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 148 T^{2} - 192 T^{3} + 9942 T^{4} - 192 p T^{5} + 148 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 208 T^{2} + 936 T^{3} + 17751 T^{4} + 936 p T^{5} + 208 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 108 T^{2} - 314 T^{3} + 3596 T^{4} - 314 p T^{5} + 108 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 130 T^{2} + 56 T^{3} + 8299 T^{4} + 56 p T^{5} + 130 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 220 T^{2} + 1014 T^{3} + 20916 T^{4} + 1014 p T^{5} + 220 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 34 T + 708 T^{2} + 9614 T^{3} + 96764 T^{4} + 9614 p T^{5} + 708 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 280 T^{2} + 1584 T^{3} + 8754 T^{4} + 1584 p T^{5} + 280 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 260 T^{2} + 1188 T^{3} + 30039 T^{4} + 1188 p T^{5} + 260 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 18 T + 352 T^{2} - 4068 T^{3} + 48255 T^{4} - 4068 p T^{5} + 352 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 4 p T^{2} + 2758 T^{3} + 56428 T^{4} + 2758 p T^{5} + 4 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
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

−6.89256741253206980856122192751, −6.80287284151893631613063728041, −6.49406329774714710309650643330, −6.19554398658212443395261232619, −6.08156553143390660254449756866, −5.58679436920089171284389379055, −5.29470977724580972160533397798, −5.26685046975194888623364609244, −4.96647502182061011725788867182, −4.55577174044929259106250478988, −4.48346882859035183347742435896, −4.35565931273346539201532666991, −4.25270741380783287526589110743, −4.07855371753571635721268158847, −3.82623127857224069888535542959, −3.67041115647824985046631985943, −3.20913713726664230166334120555, −2.86066844248215195472565500382, −2.68020753174354977592444879940, −2.30776677554152168679952599804, −2.17200505259431307073772183364, −2.09760007356032911815456218351, −1.65207930392255562937278984603, −1.54293706312417234013246431857, −1.49319400556782165634472353133, 0, 0, 0, 0, 1.49319400556782165634472353133, 1.54293706312417234013246431857, 1.65207930392255562937278984603, 2.09760007356032911815456218351, 2.17200505259431307073772183364, 2.30776677554152168679952599804, 2.68020753174354977592444879940, 2.86066844248215195472565500382, 3.20913713726664230166334120555, 3.67041115647824985046631985943, 3.82623127857224069888535542959, 4.07855371753571635721268158847, 4.25270741380783287526589110743, 4.35565931273346539201532666991, 4.48346882859035183347742435896, 4.55577174044929259106250478988, 4.96647502182061011725788867182, 5.26685046975194888623364609244, 5.29470977724580972160533397798, 5.58679436920089171284389379055, 6.08156553143390660254449756866, 6.19554398658212443395261232619, 6.49406329774714710309650643330, 6.80287284151893631613063728041, 6.89256741253206980856122192751

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