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Properties

Label 8-9072e4-1.1-c1e4-0-6
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 $4$

Origins

Origins of factors

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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: \(4\)
Selberg data: \((8,\ 2^{16} \cdot 3^{16} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
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

−5.88537444333278283220799938994, −5.39847347473967546308232887883, −5.39474151881969597738670017715, −5.30611497389425129248376088195, −5.05941566916259113032923273288, −4.74480903477121165809661065409, −4.69690065548878623132651769027, −4.58482005864362508387109887228, −4.28803813610274511830501722288, −4.18689383892473664000184782848, −3.90378642240516445936448805422, −3.90367577276030564813426752454, −3.73699196801692099942447127034, −3.19655714663749970522190012439, −3.03686875319789679456363576559, −2.94565516525182439316079147245, −2.91287522978175840846820632931, −2.39657725775948695474256271138, −2.23248415019445111402167729080, −2.21393132171979476435624424493, −2.06068354559374787690510179628, −1.49922278782665926711301845501, −1.25690113769163489612446343164, −1.23893695926593487079762820996, −0.932170750311818498610116087451, 0, 0, 0, 0, 0.932170750311818498610116087451, 1.23893695926593487079762820996, 1.25690113769163489612446343164, 1.49922278782665926711301845501, 2.06068354559374787690510179628, 2.21393132171979476435624424493, 2.23248415019445111402167729080, 2.39657725775948695474256271138, 2.91287522978175840846820632931, 2.94565516525182439316079147245, 3.03686875319789679456363576559, 3.19655714663749970522190012439, 3.73699196801692099942447127034, 3.90367577276030564813426752454, 3.90378642240516445936448805422, 4.18689383892473664000184782848, 4.28803813610274511830501722288, 4.58482005864362508387109887228, 4.69690065548878623132651769027, 4.74480903477121165809661065409, 5.05941566916259113032923273288, 5.30611497389425129248376088195, 5.39474151881969597738670017715, 5.39847347473967546308232887883, 5.88537444333278283220799938994

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