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Properties

Label 8-2475e4-1.1-c3e4-0-0
Degree $8$
Conductor $3.752\times 10^{13}$
Sign $1$
Analytic cond. $4.54742\times 10^{8}$
Root an. cond. $12.0842$
Motivic weight $3$
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  − 2·2-s − 21·4-s + 11·7-s + 52·8-s + 44·11-s − 25·13-s − 22·14-s + 213·16-s + 85·17-s + 118·19-s − 88·22-s − 10·23-s + 50·26-s − 231·28-s − 251·29-s − 135·31-s − 630·32-s − 170·34-s − 419·37-s − 236·38-s + 103·41-s − 15·43-s − 924·44-s + 20·46-s + 665·47-s − 124·49-s + 525·52-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.62·4-s + 0.593·7-s + 2.29·8-s + 1.20·11-s − 0.533·13-s − 0.419·14-s + 3.32·16-s + 1.21·17-s + 1.42·19-s − 0.852·22-s − 0.0906·23-s + 0.377·26-s − 1.55·28-s − 1.60·29-s − 0.782·31-s − 3.48·32-s − 0.857·34-s − 1.86·37-s − 1.00·38-s + 0.392·41-s − 0.0531·43-s − 3.16·44-s + 0.0641·46-s + 2.06·47-s − 0.361·49-s + 1.40·52-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4.54742\times 10^{8}\)
Root analytic conductor: \(12.0842\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(2.250639883\)
\(L(\frac12)\) \(\approx\) \(2.250639883\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5 \( 1 \)
11$C_1$ \( ( 1 - p T )^{4} \)
good2$D_{4}$ \( ( 1 + T + 3 p^{2} T^{2} + p^{3} T^{3} + p^{6} T^{4} )^{2} \)
7$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 5 p^{2} T^{2} - 4272 T^{3} + 235846 T^{4} - 4272 p^{3} T^{5} + 5 p^{8} T^{6} - 11 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 25 T + 2848 T^{2} - 89705 T^{3} + 1153550 T^{4} - 89705 p^{3} T^{5} + 2848 p^{6} T^{6} + 25 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 5 p T + 17959 T^{2} - 63382 p T^{3} + 126876108 T^{4} - 63382 p^{4} T^{5} + 17959 p^{6} T^{6} - 5 p^{10} T^{7} + p^{12} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 118 T + 23772 T^{2} - 1971120 T^{3} + 225214933 T^{4} - 1971120 p^{3} T^{5} + 23772 p^{6} T^{6} - 118 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 19600 T^{2} - 1389080 T^{3} + 201166953 T^{4} - 1389080 p^{3} T^{5} + 19600 p^{6} T^{6} + 10 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 251 T + 64738 T^{2} + 9591241 T^{3} + 1925819610 T^{4} + 9591241 p^{3} T^{5} + 64738 p^{6} T^{6} + 251 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 135 T + 73642 T^{2} + 11799675 T^{3} + 2705005482 T^{4} + 11799675 p^{3} T^{5} + 73642 p^{6} T^{6} + 135 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 419 T + 203005 T^{2} + 49657538 T^{3} + 14552591126 T^{4} + 49657538 p^{3} T^{5} + 203005 p^{6} T^{6} + 419 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 103 T + 193405 T^{2} - 16488354 T^{3} + 18502936946 T^{4} - 16488354 p^{3} T^{5} + 193405 p^{6} T^{6} - 103 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 175040 T^{2} + 43959 T^{3} + 15986121134 T^{4} + 43959 p^{3} T^{5} + 175040 p^{6} T^{6} + 15 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 665 T + 375451 T^{2} - 147542792 T^{3} + 57422466252 T^{4} - 147542792 p^{3} T^{5} + 375451 p^{6} T^{6} - 665 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 116 T + 240584 T^{2} + 1497452 T^{3} + 30580910270 T^{4} + 1497452 p^{3} T^{5} + 240584 p^{6} T^{6} + 116 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 951 T + 746257 T^{2} + 351355448 T^{3} + 184211670670 T^{4} + 351355448 p^{3} T^{5} + 746257 p^{6} T^{6} + 951 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 350 T + 481192 T^{2} + 204604298 T^{3} + 125591896574 T^{4} + 204604298 p^{3} T^{5} + 481192 p^{6} T^{6} + 350 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 266 T + 252240 T^{2} + 289246466 T^{3} + 93419783054 T^{4} + 289246466 p^{3} T^{5} + 252240 p^{6} T^{6} + 266 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 1526 T + 2164088 T^{2} + 1770899240 T^{3} + 1299777873953 T^{4} + 1770899240 p^{3} T^{5} + 2164088 p^{6} T^{6} + 1526 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 566 T - 231032 T^{2} + 82463206 T^{3} + 143348281550 T^{4} + 82463206 p^{3} T^{5} - 231032 p^{6} T^{6} - 566 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 2567 T + 3969287 T^{2} - 4195920960 T^{3} + 3375898676528 T^{4} - 4195920960 p^{3} T^{5} + 3969287 p^{6} T^{6} - 2567 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 961 T + 1800414 T^{2} + 1174919601 T^{3} + 1350021543298 T^{4} + 1174919601 p^{3} T^{5} + 1800414 p^{6} T^{6} + 961 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 1053 T + 2481236 T^{2} + 1650076467 T^{3} + 2370212988070 T^{4} + 1650076467 p^{3} T^{5} + 2481236 p^{6} T^{6} + 1053 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 172 T + 1942834 T^{2} - 351210592 T^{3} + 1907414271035 T^{4} - 351210592 p^{3} T^{5} + 1942834 p^{6} T^{6} - 172 p^{9} T^{7} + p^{12} T^{8} \)
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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

−5.83524269859890033442914012191, −5.54437467212630243330406901198, −5.51343465476860244164874620229, −5.45127390656814556238789926711, −5.19088727651759080323949604692, −4.73459119658325639469652675809, −4.59227827103913650003644630159, −4.58253718558698904786675485275, −4.57707136351012098982930489505, −3.92908938595430507414663580807, −3.83466365936903453268888405604, −3.71973612703745306547537712424, −3.62902157620089940221784407549, −3.12359792615209602630099063474, −3.10813918028341016601204210055, −2.78282542531429569650511002225, −2.38865303475903047715748641450, −1.90396262163787836994983372872, −1.58535481268965662317127937340, −1.57723773912821338721199253284, −1.43659437056108675197357025778, −0.862883092004760457964225198231, −0.62624090710593914151369063784, −0.43957364783193636616135185777, −0.32394960496763454025358588641, 0.32394960496763454025358588641, 0.43957364783193636616135185777, 0.62624090710593914151369063784, 0.862883092004760457964225198231, 1.43659437056108675197357025778, 1.57723773912821338721199253284, 1.58535481268965662317127937340, 1.90396262163787836994983372872, 2.38865303475903047715748641450, 2.78282542531429569650511002225, 3.10813918028341016601204210055, 3.12359792615209602630099063474, 3.62902157620089940221784407549, 3.71973612703745306547537712424, 3.83466365936903453268888405604, 3.92908938595430507414663580807, 4.57707136351012098982930489505, 4.58253718558698904786675485275, 4.59227827103913650003644630159, 4.73459119658325639469652675809, 5.19088727651759080323949604692, 5.45127390656814556238789926711, 5.51343465476860244164874620229, 5.54437467212630243330406901198, 5.83524269859890033442914012191

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