Properties

Label 8-9075e4-1.1-c1e4-0-24
Degree $8$
Conductor $6.782\times 10^{15}$
Sign $1$
Analytic cond. $2.75736\times 10^{7}$
Root an. cond. $8.51259$
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-s + 4·3-s − 3·4-s + 4·6-s − 5·8-s + 10·9-s − 12·12-s + 7·13-s + 8·17-s + 10·18-s − 11·19-s − 5·23-s − 20·24-s + 7·26-s + 20·27-s − 17·29-s − 5·31-s + 9·32-s + 8·34-s − 30·36-s − 15·37-s − 11·38-s + 28·39-s − 10·41-s − 4·43-s − 5·46-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 2.30·3-s − 3/2·4-s + 1.63·6-s − 1.76·8-s + 10/3·9-s − 3.46·12-s + 1.94·13-s + 1.94·17-s + 2.35·18-s − 2.52·19-s − 1.04·23-s − 4.08·24-s + 1.37·26-s + 3.84·27-s − 3.15·29-s − 0.898·31-s + 1.59·32-s + 1.37·34-s − 5·36-s − 2.46·37-s − 1.78·38-s + 4.48·39-s − 1.56·41-s − 0.609·43-s − 0.737·46-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.75736\times 10^{7}\)
Root analytic conductor: \(8.51259\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 - T )^{4} \)
5 \( 1 \)
11 \( 1 \)
good2$C_4\times C_2$ \( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
7$((C_8 : C_2):C_2):C_2$ \( 1 + 18 T^{2} + 15 T^{3} + 149 T^{4} + 15 p T^{5} + 18 p^{2} T^{6} + p^{4} T^{8} \)
13$((C_8 : C_2):C_2):C_2$ \( 1 - 7 T + 36 T^{2} - 71 T^{3} + 239 T^{4} - 71 p T^{5} + 36 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
17$((C_8 : C_2):C_2):C_2$ \( 1 - 8 T + 62 T^{2} - 320 T^{3} + 1471 T^{4} - 320 p T^{5} + 62 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
19$((C_8 : C_2):C_2):C_2$ \( 1 + 11 T + 82 T^{2} + 443 T^{3} + 2155 T^{4} + 443 p T^{5} + 82 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
23$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 67 T^{2} + 370 T^{3} + 2019 T^{4} + 370 p T^{5} + 67 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
29$((C_8 : C_2):C_2):C_2$ \( 1 + 17 T + 200 T^{2} + 1517 T^{3} + 9499 T^{4} + 1517 p T^{5} + 200 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
31$((C_8 : C_2):C_2):C_2$ \( 1 + 5 T + 79 T^{2} + 230 T^{3} + 2971 T^{4} + 230 p T^{5} + 79 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
37$((C_8 : C_2):C_2):C_2$ \( 1 + 15 T + 213 T^{2} + 1710 T^{3} + 12869 T^{4} + 1710 p T^{5} + 213 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
41$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 154 T^{2} + 1085 T^{3} + 9261 T^{4} + 1085 p T^{5} + 154 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
43$((C_8 : C_2):C_2):C_2$ \( 1 + 4 T + 48 T^{2} - 25 T^{3} - 139 T^{4} - 25 p T^{5} + 48 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
47$((C_8 : C_2):C_2):C_2$ \( 1 - 8 T + 172 T^{2} - 985 T^{3} + 11871 T^{4} - 985 p T^{5} + 172 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
53$((C_8 : C_2):C_2):C_2$ \( 1 + 10 T + 202 T^{2} + 1565 T^{3} + 15819 T^{4} + 1565 p T^{5} + 202 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
59$((C_8 : C_2):C_2):C_2$ \( 1 - 9 T + 167 T^{2} - 1332 T^{3} + 14085 T^{4} - 1332 p T^{5} + 167 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
61$((C_8 : C_2):C_2):C_2$ \( 1 + 37 T + 748 T^{2} + 9769 T^{3} + 90385 T^{4} + 9769 p T^{5} + 748 p^{2} T^{6} + 37 p^{3} T^{7} + p^{4} T^{8} \)
67$((C_8 : C_2):C_2):C_2$ \( 1 + 3 T + 222 T^{2} + 675 T^{3} + 20801 T^{4} + 675 p T^{5} + 222 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
71$((C_8 : C_2):C_2):C_2$ \( 1 - 13 T + 323 T^{2} - 2686 T^{3} + 35395 T^{4} - 2686 p T^{5} + 323 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
73$((C_8 : C_2):C_2):C_2$ \( 1 - 15 T + 252 T^{2} - 1965 T^{3} + 21929 T^{4} - 1965 p T^{5} + 252 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
79$((C_8 : C_2):C_2):C_2$ \( 1 + 20 T + 411 T^{2} + 4750 T^{3} + 52151 T^{4} + 4750 p T^{5} + 411 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
83$((C_8 : C_2):C_2):C_2$ \( 1 + 17 T + 71 T^{2} - 1114 T^{3} - 16661 T^{4} - 1114 p T^{5} + 71 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
89$((C_8 : C_2):C_2):C_2$ \( 1 + 24 T + 227 T^{2} + 342 T^{3} - 7305 T^{4} + 342 p T^{5} + 227 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
97$((C_8 : C_2):C_2):C_2$ \( 1 - 32 T + 672 T^{2} - 9775 T^{3} + 111611 T^{4} - 9775 p T^{5} + 672 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.76055646940915180166495840466, −5.38520148406695710152936422409, −5.36836886445807676523364208375, −5.26015502179927640365554409609, −5.19782837395241635248582351180, −4.60036928240713583477992620519, −4.46346390981923622218701038323, −4.43099268772835460908141286315, −4.40756419862776708429556236097, −4.00443957730519471152997071796, −3.84660818589328531445829874461, −3.63603918110538956297332205068, −3.63245477403164625040941187143, −3.39900399208596157288518477454, −3.36315584209394516025216254609, −3.18730180634712181132733558817, −2.76706745227759346555886894179, −2.63627930257226483940601355060, −2.39804246008977936276438104262, −2.04918830718867368150424995692, −1.82784734026728706459808724839, −1.71542902122574543358975602552, −1.33334431396348019289872616618, −1.30850266896252076702205213789, −1.27646371447734900331934157143, 0, 0, 0, 0, 1.27646371447734900331934157143, 1.30850266896252076702205213789, 1.33334431396348019289872616618, 1.71542902122574543358975602552, 1.82784734026728706459808724839, 2.04918830718867368150424995692, 2.39804246008977936276438104262, 2.63627930257226483940601355060, 2.76706745227759346555886894179, 3.18730180634712181132733558817, 3.36315584209394516025216254609, 3.39900399208596157288518477454, 3.63245477403164625040941187143, 3.63603918110538956297332205068, 3.84660818589328531445829874461, 4.00443957730519471152997071796, 4.40756419862776708429556236097, 4.43099268772835460908141286315, 4.46346390981923622218701038323, 4.60036928240713583477992620519, 5.19782837395241635248582351180, 5.26015502179927640365554409609, 5.36836886445807676523364208375, 5.38520148406695710152936422409, 5.76055646940915180166495840466

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