Properties

Label 8-4016e4-1.1-c1e4-0-0
Degree $8$
Conductor $2.601\times 10^{14}$
Sign $1$
Analytic cond. $1.05750\times 10^{6}$
Root an. cond. $5.66285$
Motivic weight $1$
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  + 2·3-s − 3·5-s + 3·7-s − 6·9-s + 3·11-s − 12·13-s − 6·15-s + 17-s + 9·19-s + 6·21-s − 4·23-s − 9·25-s − 17·27-s − 12·29-s + 2·31-s + 6·33-s − 9·35-s − 13·37-s − 24·39-s + 41-s + 5·43-s + 18·45-s − 12·47-s − 14·49-s + 2·51-s + 5·53-s − 9·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.34·5-s + 1.13·7-s − 2·9-s + 0.904·11-s − 3.32·13-s − 1.54·15-s + 0.242·17-s + 2.06·19-s + 1.30·21-s − 0.834·23-s − 9/5·25-s − 3.27·27-s − 2.22·29-s + 0.359·31-s + 1.04·33-s − 1.52·35-s − 2.13·37-s − 3.84·39-s + 0.156·41-s + 0.762·43-s + 2.68·45-s − 1.75·47-s − 2·49-s + 0.280·51-s + 0.686·53-s − 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 251^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 251^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 251^{4}\)
Sign: $1$
Analytic conductor: \(1.05750\times 10^{6}\)
Root analytic conductor: \(5.66285\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 251^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad2 \( 1 \)
251$C_1$ \( ( 1 - T )^{4} \)
good3$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 10 T^{2} - 5 p T^{3} + 43 T^{4} - 5 p^{2} T^{5} + 10 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 18 T^{2} + 43 T^{3} + 131 T^{4} + 43 p T^{5} + 18 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 23 T^{2} - 44 T^{3} + 213 T^{4} - 44 p T^{5} + 23 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 4 p T^{2} - 95 T^{3} + 725 T^{4} - 95 p T^{5} + 4 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 100 T^{2} + 545 T^{3} + 2303 T^{4} + 545 p T^{5} + 100 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - T + 44 T^{2} - p T^{3} + 949 T^{4} - p^{2} T^{5} + 44 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 73 T^{2} - 424 T^{3} + 2153 T^{4} - 424 p T^{5} + 73 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 79 T^{2} + 242 T^{3} + 2587 T^{4} + 242 p T^{5} + 79 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 115 T^{2} + 822 T^{3} + 5299 T^{4} + 822 p T^{5} + 115 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 74 T^{2} - 135 T^{3} + 2655 T^{4} - 135 p T^{5} + 74 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 176 T^{2} + 1251 T^{3} + 9627 T^{4} + 1251 p T^{5} + 176 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - T + 129 T^{2} + 4 T^{3} + 7095 T^{4} + 4 p T^{5} + 129 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 38 T^{2} + 65 T^{3} + 1009 T^{4} + 65 p T^{5} + 38 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 222 T^{2} + 1705 T^{3} + 16451 T^{4} + 1705 p T^{5} + 222 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 188 T^{2} - 715 T^{3} + 14449 T^{4} - 715 p T^{5} + 188 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 173 T^{2} + 946 T^{3} + 14003 T^{4} + 946 p T^{5} + 173 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 21 T + 401 T^{2} + 4332 T^{3} + 42001 T^{4} + 4332 p T^{5} + 401 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 17 T + 242 T^{2} + 2035 T^{3} + 19291 T^{4} + 2035 p T^{5} + 242 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 210 T^{2} - 1635 T^{3} + 22127 T^{4} - 1635 p T^{5} + 210 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 162 T^{2} + 697 T^{3} + 13533 T^{4} + 697 p T^{5} + 162 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 21 T + 296 T^{2} - 2355 T^{3} + 21235 T^{4} - 2355 p T^{5} + 296 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - T + 122 T^{2} - 297 T^{3} + 12743 T^{4} - 297 p T^{5} + 122 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 257 T^{2} - 1080 T^{3} + 32393 T^{4} - 1080 p T^{5} + 257 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 308 T^{2} - 1804 T^{3} + 41253 T^{4} - 1804 p T^{5} + 308 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.49122313229901411181494629090, −5.91618671173155650228961410668, −5.89454142840686474620593258291, −5.72230420754594599714992846861, −5.66546635112136242165937627083, −5.21125902035441335408612697128, −5.10920643097520626875280277379, −5.02583980592419738013725831864, −4.87520232488655626170611706791, −4.59820124719456037060083160995, −4.27184499537302090146500618881, −4.19874751532140630879447366421, −3.87603489695841317312055477176, −3.49239438361837207330312895902, −3.40746323346185897396215724909, −3.35786880508558534559026705260, −3.31019256532059595747955686555, −2.81328076116587888130456133165, −2.58698286278550430157717957730, −2.38237279249593738772883710235, −2.20916298962390257648505168281, −1.89058796809487042177090097611, −1.75149982575177187027476539056, −1.31857163001396991362419935737, −1.15120338923937659355245831759, 0, 0, 0, 0, 1.15120338923937659355245831759, 1.31857163001396991362419935737, 1.75149982575177187027476539056, 1.89058796809487042177090097611, 2.20916298962390257648505168281, 2.38237279249593738772883710235, 2.58698286278550430157717957730, 2.81328076116587888130456133165, 3.31019256532059595747955686555, 3.35786880508558534559026705260, 3.40746323346185897396215724909, 3.49239438361837207330312895902, 3.87603489695841317312055477176, 4.19874751532140630879447366421, 4.27184499537302090146500618881, 4.59820124719456037060083160995, 4.87520232488655626170611706791, 5.02583980592419738013725831864, 5.10920643097520626875280277379, 5.21125902035441335408612697128, 5.66546635112136242165937627083, 5.72230420754594599714992846861, 5.89454142840686474620593258291, 5.91618671173155650228961410668, 6.49122313229901411181494629090

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