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Properties

Label 8-33e8-1.1-c3e4-0-6
Degree $8$
Conductor $1.406\times 10^{12}$
Sign $1$
Analytic cond. $1.70441\times 10^{7}$
Root an. cond. $8.01580$
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  + 3·2-s − 13·4-s − 12·5-s − 11·7-s − 42·8-s − 36·10-s − 182·13-s − 33·14-s + 126·16-s + 57·17-s − 173·19-s + 156·20-s + 42·23-s − 271·25-s − 546·26-s + 143·28-s + 651·29-s − 170·31-s + 318·32-s + 171·34-s + 132·35-s + 244·37-s − 519·38-s + 504·40-s − 102·41-s − 322·43-s + 126·46-s + ⋯
L(s)  = 1  + 1.06·2-s − 1.62·4-s − 1.07·5-s − 0.593·7-s − 1.85·8-s − 1.13·10-s − 3.88·13-s − 0.629·14-s + 1.96·16-s + 0.813·17-s − 2.08·19-s + 1.74·20-s + 0.380·23-s − 2.16·25-s − 4.11·26-s + 0.965·28-s + 4.16·29-s − 0.984·31-s + 1.75·32-s + 0.862·34-s + 0.637·35-s + 1.08·37-s − 2.21·38-s + 1.99·40-s − 0.388·41-s − 1.14·43-s + 0.403·46-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.70441\times 10^{7}\)
Root analytic conductor: \(8.01580\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{8} \cdot 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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)$
bad3 \( 1 \)
11 \( 1 \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 11 p T^{2} - 63 T^{3} + 223 T^{4} - 63 p^{3} T^{5} + 11 p^{7} T^{6} - 3 p^{9} T^{7} + p^{12} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 83 p T^{2} + 2994 T^{3} + 67819 T^{4} + 2994 p^{3} T^{5} + 83 p^{7} T^{6} + 12 p^{9} T^{7} + p^{12} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 402 T^{2} + 946 p T^{3} + 162773 T^{4} + 946 p^{4} T^{5} + 402 p^{6} T^{6} + 11 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 14 p T + 16551 T^{2} + 1015624 T^{3} + 50687192 T^{4} + 1015624 p^{3} T^{5} + 16551 p^{6} T^{6} + 14 p^{10} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 57 T + 16021 T^{2} - 707631 T^{3} + 113209144 T^{4} - 707631 p^{3} T^{5} + 16021 p^{6} T^{6} - 57 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 173 T + 24849 T^{2} + 2173159 T^{3} + 197856056 T^{4} + 2173159 p^{3} T^{5} + 24849 p^{6} T^{6} + 173 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 42 T + 28427 T^{2} + 413280 T^{3} + 349199196 T^{4} + 413280 p^{3} T^{5} + 28427 p^{6} T^{6} - 42 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 651 T + 250093 T^{2} - 62670981 T^{3} + 11524701508 T^{4} - 62670981 p^{3} T^{5} + 250093 p^{6} T^{6} - 651 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 170 T + 122685 T^{2} + 14936530 T^{3} + 5536501547 T^{4} + 14936530 p^{3} T^{5} + 122685 p^{6} T^{6} + 170 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 244 T + 107679 T^{2} - 22139072 T^{3} + 5635850852 T^{4} - 22139072 p^{3} T^{5} + 107679 p^{6} T^{6} - 244 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 102 T + 111067 T^{2} - 1992132 T^{3} + 5701306888 T^{4} - 1992132 p^{3} T^{5} + 111067 p^{6} T^{6} + 102 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 322 T + 181243 T^{2} + 48993496 T^{3} + 21697488148 T^{4} + 48993496 p^{3} T^{5} + 181243 p^{6} T^{6} + 322 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 633 T + 461545 T^{2} - 166763625 T^{3} + 69749378908 T^{4} - 166763625 p^{3} T^{5} + 461545 p^{6} T^{6} - 633 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 468 T + 605161 T^{2} - 196671456 T^{3} + 135334830097 T^{4} - 196671456 p^{3} T^{5} + 605161 p^{6} T^{6} - 468 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 996 T + 928327 T^{2} - 561140838 T^{3} + 297148831615 T^{4} - 561140838 p^{3} T^{5} + 928327 p^{6} T^{6} - 996 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 413 T + 814107 T^{2} + 251879257 T^{3} + 270541264808 T^{4} + 251879257 p^{3} T^{5} + 814107 p^{6} T^{6} + 413 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 259 T + 1117027 T^{2} + 221089363 T^{3} + 492802725568 T^{4} + 221089363 p^{3} T^{5} + 1117027 p^{6} T^{6} + 259 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 237 T + 153589 T^{2} + 181426365 T^{3} + 211089307840 T^{4} + 181426365 p^{3} T^{5} + 153589 p^{6} T^{6} + 237 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 2309 T + 3335073 T^{2} + 44161957 p T^{3} + 2344749952988 T^{4} + 44161957 p^{4} T^{5} + 3335073 p^{6} T^{6} + 2309 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 2045 T + 3122832 T^{2} + 3169328350 T^{3} + 2582596622363 T^{4} + 3169328350 p^{3} T^{5} + 3122832 p^{6} T^{6} + 2045 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 639 T + 1399570 T^{2} - 960059370 T^{3} + 1037473710253 T^{4} - 960059370 p^{3} T^{5} + 1399570 p^{6} T^{6} - 639 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 894 T + 1324211 T^{2} + 258930180 T^{3} + 627382282200 T^{4} + 258930180 p^{3} T^{5} + 1324211 p^{6} T^{6} + 894 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 1432 T + 4102131 T^{2} - 3836622086 T^{3} + 5790534104459 T^{4} - 3836622086 p^{3} T^{5} + 4102131 p^{6} T^{6} - 1432 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.11108998209965616009716787295, −7.08398251202815204101921355009, −6.44742741851502589454857043380, −6.41765048404704196777523415633, −6.34266211177854930926179490606, −5.68562270541559398271640777060, −5.67557527250201902834021113454, −5.41242857838223352003242089528, −5.30586901259035794277122308811, −4.81301523918779281538838974978, −4.63171120650681052661520753290, −4.62093904740747766656411176384, −4.53678711212230964350529364155, −4.07464670482767420858942262825, −4.02726040380433144101719871332, −3.79514674089528841354725191393, −3.60008652712712346178447029254, −2.95394121583600514740781409205, −2.79593947776054368115292217111, −2.73015652540452740231132534677, −2.44731382669805019263945805908, −2.05017024369882153885271691483, −1.57527599287529073637425854119, −1.10795422529832146280721974251, −0.844654078561291294092703979951, 0, 0, 0, 0, 0.844654078561291294092703979951, 1.10795422529832146280721974251, 1.57527599287529073637425854119, 2.05017024369882153885271691483, 2.44731382669805019263945805908, 2.73015652540452740231132534677, 2.79593947776054368115292217111, 2.95394121583600514740781409205, 3.60008652712712346178447029254, 3.79514674089528841354725191393, 4.02726040380433144101719871332, 4.07464670482767420858942262825, 4.53678711212230964350529364155, 4.62093904740747766656411176384, 4.63171120650681052661520753290, 4.81301523918779281538838974978, 5.30586901259035794277122308811, 5.41242857838223352003242089528, 5.67557527250201902834021113454, 5.68562270541559398271640777060, 6.34266211177854930926179490606, 6.41765048404704196777523415633, 6.44742741851502589454857043380, 7.08398251202815204101921355009, 7.11108998209965616009716787295

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