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Properties

Label 8-9072e4-1.1-c1e4-0-3
Degree $8$
Conductor $6.773\times 10^{15}$
Sign $1$
Analytic cond. $2.75372\times 10^{7}$
Root an. cond. $8.51118$
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  + 3·5-s + 4·7-s + 7·11-s − 3·13-s + 3·17-s + 4·19-s + 2·23-s − 3·25-s + 9·29-s + 3·31-s + 12·35-s + 3·37-s − 9·41-s + 8·43-s + 3·47-s + 10·49-s + 6·53-s + 21·55-s + 10·59-s − 20·61-s − 9·65-s + 11·67-s + 3·71-s − 24·73-s + 28·77-s + 21·79-s + 8·83-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.51·7-s + 2.11·11-s − 0.832·13-s + 0.727·17-s + 0.917·19-s + 0.417·23-s − 3/5·25-s + 1.67·29-s + 0.538·31-s + 2.02·35-s + 0.493·37-s − 1.40·41-s + 1.21·43-s + 0.437·47-s + 10/7·49-s + 0.824·53-s + 2.83·55-s + 1.30·59-s − 2.56·61-s − 1.11·65-s + 1.34·67-s + 0.356·71-s − 2.80·73-s + 3.19·77-s + 2.36·79-s + 0.878·83-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{16} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2.75372\times 10^{7}\)
Root analytic conductor: \(8.51118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{16} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(22.59569539\)
\(L(\frac12)\) \(\approx\) \(22.59569539\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
good5$C_2^3:S_4$ \( 1 - 3 T + 12 T^{2} - 36 T^{3} + 68 T^{4} - 36 p T^{5} + 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 7 T + 51 T^{2} - 219 T^{3} + 865 T^{4} - 219 p T^{5} + 51 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 3 T + 41 T^{2} + 90 T^{3} + 738 T^{4} + 90 p T^{5} + 41 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 3 T + 41 T^{2} - 189 T^{3} + 807 T^{4} - 189 p T^{5} + 41 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 4 T + 49 T^{2} - 193 T^{3} + 1297 T^{4} - 193 p T^{5} + 49 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 2 T + 63 T^{2} - 159 T^{3} + 1876 T^{4} - 159 p T^{5} + 63 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 9 T + 68 T^{2} - 270 T^{3} + 1290 T^{4} - 270 p T^{5} + 68 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 3 T + 50 T^{2} - 252 T^{3} + 1872 T^{4} - 252 p T^{5} + 50 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 3 T + 140 T^{2} - 324 T^{3} + 7620 T^{4} - 324 p T^{5} + 140 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 9 T + 117 T^{2} + 717 T^{3} + 6341 T^{4} + 717 p T^{5} + 117 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 8 T + 137 T^{2} - 1005 T^{3} + 8087 T^{4} - 1005 p T^{5} + 137 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 3 T + 150 T^{2} - 468 T^{3} + 9674 T^{4} - 468 p T^{5} + 150 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 6 T + 59 T^{2} - 621 T^{3} + 4704 T^{4} - 621 p T^{5} + 59 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 10 T + 171 T^{2} - 1623 T^{3} + 13357 T^{4} - 1623 p T^{5} + 171 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 20 T + 239 T^{2} + 2109 T^{3} + 15872 T^{4} + 2109 p T^{5} + 239 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 11 T + 203 T^{2} - 1461 T^{3} + 17099 T^{4} - 1461 p T^{5} + 203 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 3 T + 276 T^{2} - 630 T^{3} + 29126 T^{4} - 630 p T^{5} + 276 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 24 T + 425 T^{2} + 4923 T^{3} + 48495 T^{4} + 4923 p T^{5} + 425 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 21 T + 308 T^{2} - 3360 T^{3} + 34200 T^{4} - 3360 p T^{5} + 308 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 8 T + 323 T^{2} - 1865 T^{3} + 39832 T^{4} - 1865 p T^{5} + 323 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 6 T + 263 T^{2} - 1377 T^{3} + 32646 T^{4} - 1377 p T^{5} + 263 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 16 T + 431 T^{2} + 4491 T^{3} + 64415 T^{4} + 4491 p T^{5} + 431 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.53436437482985723931070541640, −5.13123673670493237613751751424, −5.04234221229493937163527486949, −4.87669984156020332066751490579, −4.83695168485013488034102382981, −4.43030818379174402772418358871, −4.32512768808615856176959528598, −4.25716991660781829846514590230, −4.08716375355915246043587342769, −3.54831750646384400438800769796, −3.52114855793234702749686572620, −3.50433801617072397459856642714, −3.29265262190363711714625792426, −2.70003165087804582272967238649, −2.62139302951729572980904617025, −2.50684949439012448178379908291, −2.49433738744905614253484444343, −1.82284759389293295640721178836, −1.79778890312184061659489112914, −1.66373333869864534560562723323, −1.53946745654094549059572115979, −1.08455478689994092673817065389, −0.913330931055191134098597923930, −0.73682861972628319489904020219, −0.42958412167668084649153645195, 0.42958412167668084649153645195, 0.73682861972628319489904020219, 0.913330931055191134098597923930, 1.08455478689994092673817065389, 1.53946745654094549059572115979, 1.66373333869864534560562723323, 1.79778890312184061659489112914, 1.82284759389293295640721178836, 2.49433738744905614253484444343, 2.50684949439012448178379908291, 2.62139302951729572980904617025, 2.70003165087804582272967238649, 3.29265262190363711714625792426, 3.50433801617072397459856642714, 3.52114855793234702749686572620, 3.54831750646384400438800769796, 4.08716375355915246043587342769, 4.25716991660781829846514590230, 4.32512768808615856176959528598, 4.43030818379174402772418358871, 4.83695168485013488034102382981, 4.87669984156020332066751490579, 5.04234221229493937163527486949, 5.13123673670493237613751751424, 5.53436437482985723931070541640

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