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Properties

Label 8-1400e4-1.1-c3e4-0-6
Degree $8$
Conductor $3.842\times 10^{12}$
Sign $1$
Analytic cond. $4.65560\times 10^{7}$
Root an. cond. $9.08860$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s + 28·7-s − 25·9-s + 22·11-s − 18·13-s − 153·17-s − 33·19-s − 140·21-s − 55·23-s + 74·27-s − 279·29-s + 60·31-s − 110·33-s + 99·37-s + 90·39-s − 119·41-s − 773·43-s + 176·47-s + 490·49-s + 765·51-s + 6·53-s + 165·57-s + 476·59-s + 320·61-s − 700·63-s − 42·67-s + 275·69-s + ⋯
L(s)  = 1  − 0.962·3-s + 1.51·7-s − 0.925·9-s + 0.603·11-s − 0.384·13-s − 2.18·17-s − 0.398·19-s − 1.45·21-s − 0.498·23-s + 0.527·27-s − 1.78·29-s + 0.347·31-s − 0.580·33-s + 0.439·37-s + 0.369·39-s − 0.453·41-s − 2.74·43-s + 0.546·47-s + 10/7·49-s + 2.10·51-s + 0.0155·53-s + 0.383·57-s + 1.05·59-s + 0.671·61-s − 1.39·63-s − 0.0765·67-s + 0.479·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(4.65560\times 10^{7}\)
Root analytic conductor: \(9.08860\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
5 \( 1 \)
7$C_1$ \( ( 1 - p T )^{4} \)
good3$C_2 \wr S_4$ \( 1 + 5 T + 50 T^{2} + 301 T^{3} + 550 p T^{4} + 301 p^{3} T^{5} + 50 p^{6} T^{6} + 5 p^{9} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 2 p T + 2892 T^{2} - 74728 T^{3} + 5323837 T^{4} - 74728 p^{3} T^{5} + 2892 p^{6} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 18 T + 6920 T^{2} + 121086 T^{3} + 20761038 T^{4} + 121086 p^{3} T^{5} + 6920 p^{6} T^{6} + 18 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 9 p T + 18588 T^{2} + 1599371 T^{3} + 125708470 T^{4} + 1599371 p^{3} T^{5} + 18588 p^{6} T^{6} + 9 p^{10} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 33 T + 19618 T^{2} + 356253 T^{3} + 171883154 T^{4} + 356253 p^{3} T^{5} + 19618 p^{6} T^{6} + 33 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 55 T + 17467 T^{2} + 753604 T^{3} + 291896376 T^{4} + 753604 p^{3} T^{5} + 17467 p^{6} T^{6} + 55 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 279 T + 106557 T^{2} + 18195598 T^{3} + 3909140026 T^{4} + 18195598 p^{3} T^{5} + 106557 p^{6} T^{6} + 279 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 60 T + 32272 T^{2} + 7369100 T^{3} - 313209858 T^{4} + 7369100 p^{3} T^{5} + 32272 p^{6} T^{6} - 60 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 99 T + 41807 T^{2} - 13219590 T^{3} + 1249812668 T^{4} - 13219590 p^{3} T^{5} + 41807 p^{6} T^{6} - 99 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 119 T + 225366 T^{2} + 16362381 T^{3} + 21322973506 T^{4} + 16362381 p^{3} T^{5} + 225366 p^{6} T^{6} + 119 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 + 773 T + 264683 T^{2} + 30356656 T^{3} + 219759648 T^{4} + 30356656 p^{3} T^{5} + 264683 p^{6} T^{6} + 773 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 176 T + 149508 T^{2} - 14494192 T^{3} + 9093440774 T^{4} - 14494192 p^{3} T^{5} + 149508 p^{6} T^{6} - 176 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 6 T + 449112 T^{2} + 4800494 T^{3} + 92030995966 T^{4} + 4800494 p^{3} T^{5} + 449112 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 476 T + 199856 T^{2} - 48695012 T^{3} - 22465352402 T^{4} - 48695012 p^{3} T^{5} + 199856 p^{6} T^{6} - 476 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 320 T + 795960 T^{2} - 190082312 T^{3} + 257734504718 T^{4} - 190082312 p^{3} T^{5} + 795960 p^{6} T^{6} - 320 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 42 T + 693536 T^{2} + 40603400 T^{3} + 241188284081 T^{4} + 40603400 p^{3} T^{5} + 693536 p^{6} T^{6} + 42 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 1937 T + 2340661 T^{2} - 2057674332 T^{3} + 1392254929898 T^{4} - 2057674332 p^{3} T^{5} + 2340661 p^{6} T^{6} - 1937 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 1253 T + 2097286 T^{2} + 1558907511 T^{3} + 1335145319346 T^{4} + 1558907511 p^{3} T^{5} + 2097286 p^{6} T^{6} + 1253 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 829 T + 1529273 T^{2} + 1139447956 T^{3} + 1051534839326 T^{4} + 1139447956 p^{3} T^{5} + 1529273 p^{6} T^{6} + 829 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 145 T + 658244 T^{2} - 372100355 T^{3} + 48620862046 T^{4} - 372100355 p^{3} T^{5} + 658244 p^{6} T^{6} + 145 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 641 T + 1814004 T^{2} - 592537551 T^{3} + 1466188108846 T^{4} - 592537551 p^{3} T^{5} + 1814004 p^{6} T^{6} - 641 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 38 T + 644240 T^{2} - 268458690 T^{3} + 1371617246414 T^{4} - 268458690 p^{3} T^{5} + 644240 p^{6} T^{6} - 38 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.94581353365321916150182130434, −6.50183843477263540106988602778, −6.30605371519836326651640945665, −6.24893892224053353814256186724, −6.24717763747374176019435712169, −5.59743003084286811978106689228, −5.40656845046455900559046855525, −5.39556370722359875114398300330, −5.25145893465611813097349510901, −4.96663879877801336412781292662, −4.65523318957985576482255398607, −4.55351104486833419173686590491, −4.21214594858513769819021706710, −3.90481694470307806744354222699, −3.74366267008073206306855108893, −3.68026102071902855184502353153, −3.31344079712054729855141123480, −2.70333413347848120712974567139, −2.43052111241003380109030966592, −2.35774805285601478779445619842, −2.28969531965461027519602146005, −1.76607296280364085805583533596, −1.41690331018039737104038187215, −1.23316758230491917342077759087, −1.01148806010678583346134984905, 0, 0, 0, 0, 1.01148806010678583346134984905, 1.23316758230491917342077759087, 1.41690331018039737104038187215, 1.76607296280364085805583533596, 2.28969531965461027519602146005, 2.35774805285601478779445619842, 2.43052111241003380109030966592, 2.70333413347848120712974567139, 3.31344079712054729855141123480, 3.68026102071902855184502353153, 3.74366267008073206306855108893, 3.90481694470307806744354222699, 4.21214594858513769819021706710, 4.55351104486833419173686590491, 4.65523318957985576482255398607, 4.96663879877801336412781292662, 5.25145893465611813097349510901, 5.39556370722359875114398300330, 5.40656845046455900559046855525, 5.59743003084286811978106689228, 6.24717763747374176019435712169, 6.24893892224053353814256186724, 6.30605371519836326651640945665, 6.50183843477263540106988602778, 6.94581353365321916150182130434

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