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Properties

Label 8-88e8-1.1-c1e4-0-11
Degree $8$
Conductor $3.596\times 10^{15}$
Sign $1$
Analytic cond. $1.46207\times 10^{7}$
Root an. cond. $7.86359$
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 − 5-s + 7-s − 9-s + 13-s − 2·15-s + 12·17-s − 14·19-s + 2·21-s + 2·23-s − 4·25-s − 2·27-s − 9·29-s − 11·31-s − 35-s − 13·37-s + 2·39-s + 8·41-s − 3·43-s + 45-s − 7·47-s − 14·49-s + 24·51-s − 11·53-s − 28·57-s + 10·59-s − 17·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 0.377·7-s − 1/3·9-s + 0.277·13-s − 0.516·15-s + 2.91·17-s − 3.21·19-s + 0.436·21-s + 0.417·23-s − 4/5·25-s − 0.384·27-s − 1.67·29-s − 1.97·31-s − 0.169·35-s − 2.13·37-s + 0.320·39-s + 1.24·41-s − 0.457·43-s + 0.149·45-s − 1.02·47-s − 2·49-s + 3.36·51-s − 1.51·53-s − 3.70·57-s + 1.30·59-s − 2.17·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.46207\times 10^{7}\)
Root analytic conductor: \(7.86359\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{24} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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)$
bad2 \( 1 \)
11 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 23 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + T + p T^{2} - 7 T^{3} + 4 T^{4} - 7 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - T + 15 T^{2} - 5 T^{3} + 108 T^{4} - 5 p T^{5} + 15 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - T + 29 T^{2} + 27 T^{3} + 372 T^{4} + 27 p T^{5} + 29 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 77 T^{2} - 400 T^{3} + 1841 T^{4} - 400 p T^{5} + 77 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
19$D_{4}$ \( ( 1 + 7 T + 49 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 40 T^{2} - 10 T^{3} + 718 T^{4} - 10 p T^{5} + 40 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 123 T^{2} + 739 T^{3} + 5408 T^{4} + 739 p T^{5} + 123 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 149 T^{2} + 951 T^{3} + 7080 T^{4} + 951 p T^{5} + 149 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 161 T^{2} + 1401 T^{3} + 9132 T^{4} + 1401 p T^{5} + 161 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 149 T^{2} - 940 T^{3} + 8845 T^{4} - 940 p T^{5} + 149 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 157 T^{2} + 343 T^{3} + 9788 T^{4} + 343 p T^{5} + 157 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 167 T^{2} + 855 T^{3} + 11456 T^{4} + 855 p T^{5} + 167 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 167 T^{2} + 1177 T^{3} + 11908 T^{4} + 1177 p T^{5} + 167 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
59$D_{4}$ \( ( 1 - 5 T + 93 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 17 T + 239 T^{2} + 2275 T^{3} + 19780 T^{4} + 2275 p T^{5} + 239 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 167 T^{2} + 1265 T^{3} + 13224 T^{4} + 1265 p T^{5} + 167 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 135 T^{2} - 395 T^{3} + 8372 T^{4} - 395 p T^{5} + 135 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 3 p T^{2} - 1248 T^{3} + 22097 T^{4} - 1248 p T^{5} + 3 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 201 T^{2} - 17 p T^{3} + 20120 T^{4} - 17 p^{2} T^{5} + 201 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 413 T^{2} + 5020 T^{3} + 54769 T^{4} + 5020 p T^{5} + 413 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 383 T^{2} - 2901 T^{3} + 52208 T^{4} - 2901 p T^{5} + 383 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 313 T^{2} - 716 T^{3} + 41413 T^{4} - 716 p T^{5} + 313 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.00419795052749075300015158386, −5.58916287377057344292029315968, −5.40768011869012572498424205371, −5.33597884327277846258756182704, −5.07533751944773154547330831198, −5.04149772911330686988848859955, −5.01964392024944714027290222693, −4.43434549115001583225594166985, −4.34706378145405506202354047761, −4.03043153724563071846423083000, −3.91163002455554027261098873091, −3.81146767488701036916010999453, −3.64661769182136689775693312603, −3.45466758912784578411727256043, −3.27519272399542081116595383878, −2.94168615343972553607254177854, −2.93533527124344226005215434844, −2.49751658349611970227930018936, −2.36598514247802889379087020532, −2.33528649552230250890440698441, −1.86605424390476911485344126984, −1.59338661262053273326253443549, −1.34630834462684979551846339057, −1.29334271708865386246467739913, −1.26177916372467656207428031798, 0, 0, 0, 0, 1.26177916372467656207428031798, 1.29334271708865386246467739913, 1.34630834462684979551846339057, 1.59338661262053273326253443549, 1.86605424390476911485344126984, 2.33528649552230250890440698441, 2.36598514247802889379087020532, 2.49751658349611970227930018936, 2.93533527124344226005215434844, 2.94168615343972553607254177854, 3.27519272399542081116595383878, 3.45466758912784578411727256043, 3.64661769182136689775693312603, 3.81146767488701036916010999453, 3.91163002455554027261098873091, 4.03043153724563071846423083000, 4.34706378145405506202354047761, 4.43434549115001583225594166985, 5.01964392024944714027290222693, 5.04149772911330686988848859955, 5.07533751944773154547330831198, 5.33597884327277846258756182704, 5.40768011869012572498424205371, 5.58916287377057344292029315968, 6.00419795052749075300015158386

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