Properties

Label 8-7728e4-1.1-c1e4-0-5
Degree $8$
Conductor $3.567\times 10^{15}$
Sign $1$
Analytic cond. $1.45002\times 10^{7}$
Root an. cond. $7.85546$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 3·5-s − 4·7-s + 10·9-s − 2·11-s + 13-s − 12·15-s − 8·17-s + 2·19-s − 16·21-s + 4·23-s − 6·25-s + 20·27-s − 10·29-s + 10·31-s − 8·33-s + 12·35-s + 4·39-s − 8·41-s − 13·43-s − 30·45-s + 6·47-s + 10·49-s − 32·51-s + 5·53-s + 6·55-s + 8·57-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.34·5-s − 1.51·7-s + 10/3·9-s − 0.603·11-s + 0.277·13-s − 3.09·15-s − 1.94·17-s + 0.458·19-s − 3.49·21-s + 0.834·23-s − 6/5·25-s + 3.84·27-s − 1.85·29-s + 1.79·31-s − 1.39·33-s + 2.02·35-s + 0.640·39-s − 1.24·41-s − 1.98·43-s − 4.47·45-s + 0.875·47-s + 10/7·49-s − 4.48·51-s + 0.686·53-s + 0.809·55-s + 1.05·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1.45002\times 10^{7}\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
3$C_1$ \( ( 1 - T )^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
23$C_1$ \( ( 1 - T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 3 p T^{2} + 41 T^{3} + 104 T^{4} + 41 p T^{5} + 3 p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 29 T^{2} + 30 T^{3} + 380 T^{4} + 30 p T^{5} + 29 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - T + 23 T^{2} - 15 T^{3} + 336 T^{4} - 15 p T^{5} + 23 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 71 T^{2} + 316 T^{3} + 1672 T^{4} + 316 p T^{5} + 71 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 61 T^{2} - 78 T^{3} + 1580 T^{4} - 78 p T^{5} + 61 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 57 T^{2} + 450 T^{3} + 3068 T^{4} + 450 p T^{5} + 57 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 65 T^{2} - 450 T^{3} + 3132 T^{4} - 450 p T^{5} + 65 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 107 T^{2} - 120 T^{3} + 5104 T^{4} - 120 p T^{5} + 107 p^{2} T^{6} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 167 T^{2} + 972 T^{3} + 10328 T^{4} + 972 p T^{5} + 167 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 13 T + 135 T^{2} + 1265 T^{3} + 9048 T^{4} + 1265 p T^{5} + 135 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 120 T^{2} - 590 T^{3} + 7118 T^{4} - 590 p T^{5} + 120 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 73 T^{2} - 295 T^{3} + 6004 T^{4} - 295 p T^{5} + 73 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - T + 161 T^{2} - 305 T^{3} + 12340 T^{4} - 305 p T^{5} + 161 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 53 T^{2} + 25 T^{3} - 2052 T^{4} + 25 p T^{5} + 53 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 197 T^{2} + 415 T^{3} + 17436 T^{4} + 415 p T^{5} + 197 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 5 T + 173 T^{2} + 1345 T^{3} + 14548 T^{4} + 1345 p T^{5} + 173 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 347 T^{2} + 3680 T^{3} + 37104 T^{4} + 3680 p T^{5} + 347 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 10 T + 257 T^{2} + 1890 T^{3} + 29148 T^{4} + 1890 p T^{5} + 257 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 18 T + 357 T^{2} + 4038 T^{3} + 45948 T^{4} + 4038 p T^{5} + 357 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 31 T + 513 T^{2} + 5541 T^{3} + 53108 T^{4} + 5541 p T^{5} + 513 p^{2} T^{6} + 31 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 443 T^{2} + 5120 T^{3} + 64224 T^{4} + 5120 p T^{5} + 443 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.00035724978743530936790746270, −5.57187220425158857894489281674, −5.42268859005913717433808459723, −5.41175101172213511998256277433, −5.20131257236212580212902161350, −4.81214804430731404100551257584, −4.60221215370727771804884439155, −4.43846301463721440147027728352, −4.33833982914250271876476406274, −3.94739389278496644838021107248, −3.91143905421051222283407749834, −3.80700374528165022178462090810, −3.76255089402131327106942529173, −3.46663249532224866881665705833, −3.12737881616901594223540217925, −2.97593440308285178608419789218, −2.86020545742159243800374438650, −2.56488344005516336837722214407, −2.49217517576195735640266702170, −2.37065408272788500202491933588, −2.12878789994202413107149587426, −1.43848535218019959271765267053, −1.42337354704132020736888631440, −1.41482446602480954712250837880, −1.09889441847995240216728185159, 0, 0, 0, 0, 1.09889441847995240216728185159, 1.41482446602480954712250837880, 1.42337354704132020736888631440, 1.43848535218019959271765267053, 2.12878789994202413107149587426, 2.37065408272788500202491933588, 2.49217517576195735640266702170, 2.56488344005516336837722214407, 2.86020545742159243800374438650, 2.97593440308285178608419789218, 3.12737881616901594223540217925, 3.46663249532224866881665705833, 3.76255089402131327106942529173, 3.80700374528165022178462090810, 3.91143905421051222283407749834, 3.94739389278496644838021107248, 4.33833982914250271876476406274, 4.43846301463721440147027728352, 4.60221215370727771804884439155, 4.81214804430731404100551257584, 5.20131257236212580212902161350, 5.41175101172213511998256277433, 5.42268859005913717433808459723, 5.57187220425158857894489281674, 6.00035724978743530936790746270

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