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Properties

Label 8-9680e4-1.1-c1e4-0-0
Degree $8$
Conductor $8.780\times 10^{15}$
Sign $1$
Analytic cond. $3.56952\times 10^{7}$
Root an. cond. $8.79176$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 3·7-s − 6·9-s − 13-s + 17-s + 20·19-s − 5·23-s + 10·25-s + 5·27-s − 12·29-s + 5·31-s − 12·35-s + 7·37-s − 11·41-s + 19·43-s + 24·45-s − 5·47-s − 8·49-s − 11·53-s − 9·59-s − 12·61-s − 18·63-s + 4·65-s + 19·67-s − 5·71-s − 11·73-s + 34·79-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.13·7-s − 2·9-s − 0.277·13-s + 0.242·17-s + 4.58·19-s − 1.04·23-s + 2·25-s + 0.962·27-s − 2.22·29-s + 0.898·31-s − 2.02·35-s + 1.15·37-s − 1.71·41-s + 2.89·43-s + 3.57·45-s − 0.729·47-s − 8/7·49-s − 1.51·53-s − 1.17·59-s − 1.53·61-s − 2.26·63-s + 0.496·65-s + 2.32·67-s − 0.593·71-s − 1.28·73-s + 3.82·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(3.56952\times 10^{7}\)
Root analytic conductor: \(8.79176\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(1.346387133\)
\(L(\frac12)\) \(\approx\) \(1.346387133\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{4} \)
11 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 p T^{2} - 5 T^{3} + 17 T^{4} - 5 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 17 T^{2} - 40 T^{3} + 171 T^{4} - 40 p T^{5} + 17 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + T + 27 T^{2} + 32 T^{3} + 503 T^{4} + 32 p T^{5} + 27 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - T + 48 T^{2} - 19 T^{3} + 1073 T^{4} - 19 p T^{5} + 48 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 206 T^{2} - 1415 T^{3} + 7131 T^{4} - 1415 p T^{5} + 206 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 96 T^{2} + 335 T^{3} + 3347 T^{4} + 335 p T^{5} + 96 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 136 T^{2} + 873 T^{3} + 5755 T^{4} + 873 p T^{5} + 136 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 49 T^{2} - 340 T^{3} + 1741 T^{4} - 340 p T^{5} + 49 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 48 T^{2} + 49 T^{3} - 337 T^{4} + 49 p T^{5} + 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 170 T^{2} + 1179 T^{3} + 10259 T^{4} + 1179 p T^{5} + 170 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 19 T + 293 T^{2} - 2740 T^{3} + 21711 T^{4} - 2740 p T^{5} + 293 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 167 T^{2} + 640 T^{3} + 11449 T^{4} + 640 p T^{5} + 167 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 169 T^{2} + 1438 T^{3} + 13237 T^{4} + 1438 p T^{5} + 169 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 163 T^{2} + 1044 T^{3} + 11443 T^{4} + 1044 p T^{5} + 163 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 267 T^{2} + 2118 T^{3} + 24963 T^{4} + 2118 p T^{5} + 267 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 19 T + 290 T^{2} - 2805 T^{3} + 25803 T^{4} - 2805 p T^{5} + 290 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 238 T^{2} + 895 T^{3} + 23583 T^{4} + 895 p T^{5} + 238 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 302 T^{2} + 2397 T^{3} + 33423 T^{4} + 2397 p T^{5} + 302 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 34 T + 652 T^{2} - 8357 T^{3} + 83755 T^{4} - 8357 p T^{5} + 652 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 233 T^{2} + 1500 T^{3} + 23201 T^{4} + 1500 p T^{5} + 233 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 254 T^{2} + 1664 T^{3} + 31231 T^{4} + 1664 p T^{5} + 254 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 598 T^{2} - 8416 T^{3} + 94183 T^{4} - 8416 p T^{5} + 598 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.38559228839049551820960836687, −5.15727237969361315562847790670, −4.99432621925694533229005367638, −4.94241225488230705133492464330, −4.70285813562278213383167687716, −4.49120233507816627122435644212, −4.24947506125265566757071907338, −4.19500899658707551557245390840, −3.85621773570162230605040855742, −3.50503338915151071969107389822, −3.48995627903415740796845139834, −3.35743077262298665288595554634, −3.25882478556172795917413210906, −2.86840009337093447738587200508, −2.81986066440711716164405209266, −2.64943019945693075814562586483, −2.53076350969143971197840488204, −1.83370724456034993769278985754, −1.79234879653079756343672893153, −1.71850645042194883932187001573, −1.31132945221320451452275439074, −0.917235500499891970474969098608, −0.78423279656584149291302740495, −0.58927672982790413303063303271, −0.16185084909528918749016138006, 0.16185084909528918749016138006, 0.58927672982790413303063303271, 0.78423279656584149291302740495, 0.917235500499891970474969098608, 1.31132945221320451452275439074, 1.71850645042194883932187001573, 1.79234879653079756343672893153, 1.83370724456034993769278985754, 2.53076350969143971197840488204, 2.64943019945693075814562586483, 2.81986066440711716164405209266, 2.86840009337093447738587200508, 3.25882478556172795917413210906, 3.35743077262298665288595554634, 3.48995627903415740796845139834, 3.50503338915151071969107389822, 3.85621773570162230605040855742, 4.19500899658707551557245390840, 4.24947506125265566757071907338, 4.49120233507816627122435644212, 4.70285813562278213383167687716, 4.94241225488230705133492464330, 4.99432621925694533229005367638, 5.15727237969361315562847790670, 5.38559228839049551820960836687

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