Properties

Label 8-4840e4-1.1-c1e4-0-2
Degree $8$
Conductor $5.488\times 10^{14}$
Sign $1$
Analytic cond. $2.23095\times 10^{6}$
Root an. cond. $6.21671$
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  − 2·3-s + 4·5-s − 7·7-s − 6·9-s + 3·13-s − 8·15-s + 11·17-s + 2·19-s + 14·21-s − 11·23-s + 10·25-s + 17·27-s + 4·29-s − 17·31-s − 28·35-s − 3·37-s − 6·39-s + 13·41-s − 7·43-s − 24·45-s − 47-s + 8·49-s − 22·51-s − 15·53-s − 4·57-s − 17·59-s + 4·61-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s − 2.64·7-s − 2·9-s + 0.832·13-s − 2.06·15-s + 2.66·17-s + 0.458·19-s + 3.05·21-s − 2.29·23-s + 2·25-s + 3.27·27-s + 0.742·29-s − 3.05·31-s − 4.73·35-s − 0.493·37-s − 0.960·39-s + 2.03·41-s − 1.06·43-s − 3.57·45-s − 0.145·47-s + 8/7·49-s − 3.08·51-s − 2.06·53-s − 0.529·57-s − 2.21·59-s + 0.512·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{12} \cdot 5^{4} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(2.23095\times 10^{6}\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{12} \cdot 5^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{4} \)
11 \( 1 \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 10 T^{2} + 5 p T^{3} + 43 T^{4} + 5 p^{2} T^{5} + 10 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + p T + 41 T^{2} + 22 p T^{3} + 477 T^{4} + 22 p^{2} T^{5} + 41 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 3 T + 31 T^{2} - 8 p T^{3} + 509 T^{4} - 8 p^{2} T^{5} + 31 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 11 T + 88 T^{2} - 499 T^{3} + 2403 T^{4} - 499 p T^{5} + 88 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 34 T^{2} - 141 T^{3} + 671 T^{4} - 141 p T^{5} + 34 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 + 11 T + 134 T^{2} + 823 T^{3} + 5137 T^{4} + 823 p T^{5} + 134 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 68 T^{2} - 379 T^{3} + 2263 T^{4} - 379 p T^{5} + 68 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 17 T + 187 T^{2} + 1518 T^{3} + 9293 T^{4} + 1518 p T^{5} + 187 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 3 T + 48 T^{2} + 309 T^{3} + 813 T^{4} + 309 p T^{5} + 48 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 13 T + 158 T^{2} - 1301 T^{3} + 10135 T^{4} - 1301 p T^{5} + 158 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 7 T + 155 T^{2} + 810 T^{3} + 9693 T^{4} + 810 p T^{5} + 155 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + T + 125 T^{2} - 100 T^{3} + 7183 T^{4} - 100 p T^{5} + 125 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 201 T^{2} + 1640 T^{3} + 14327 T^{4} + 1640 p T^{5} + 201 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 17 T + 201 T^{2} + 1738 T^{3} + 15075 T^{4} + 1738 p T^{5} + 201 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 111 T^{2} - 638 T^{3} + 6911 T^{4} - 638 p T^{5} + 111 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 176 T^{2} - 1169 T^{3} + 15037 T^{4} - 1169 p T^{5} + 176 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 15 T + 140 T^{2} + 385 T^{3} - 73 T^{4} + 385 p T^{5} + 140 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 2 p T^{2} - 901 T^{3} + 14409 T^{4} - 901 p T^{5} + 2 p^{3} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 244 T^{2} + 2161 T^{3} + 28451 T^{4} + 2161 p T^{5} + 244 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 9 T + 359 T^{2} - 2272 T^{3} + 45827 T^{4} - 2272 p T^{5} + 359 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 274 T^{2} + 24 p T^{3} + 32451 T^{4} + 24 p^{2} T^{5} + 274 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 288 T^{2} - 436 T^{3} + 39233 T^{4} - 436 p T^{5} + 288 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.14930378360452239016544511183, −5.90986479284081894543370576759, −5.89950680602723040559802874398, −5.81097568073932704657902150934, −5.66146745069945851023919856596, −5.21539557044887741030676411961, −5.19936486244070317479880685098, −5.06154639951441437271598427158, −5.04404883892028181652436424424, −4.40787320562032348517628675679, −4.38738630530687487003304686354, −3.80527778884990097102940261143, −3.76879395542341523762805375104, −3.37595490781732095499423881163, −3.37365458068758893890386199553, −3.34429622718569908729876339985, −3.19435941443248751708236777271, −2.61821444946614869678694318472, −2.49561686725426978994739439666, −2.45479722197784585808057991420, −2.28612376427139758089142897776, −1.56482899299555443109336624908, −1.34001673442656258428524432973, −1.24016304496842748810740391226, −1.17854084273705461515961225490, 0, 0, 0, 0, 1.17854084273705461515961225490, 1.24016304496842748810740391226, 1.34001673442656258428524432973, 1.56482899299555443109336624908, 2.28612376427139758089142897776, 2.45479722197784585808057991420, 2.49561686725426978994739439666, 2.61821444946614869678694318472, 3.19435941443248751708236777271, 3.34429622718569908729876339985, 3.37365458068758893890386199553, 3.37595490781732095499423881163, 3.76879395542341523762805375104, 3.80527778884990097102940261143, 4.38738630530687487003304686354, 4.40787320562032348517628675679, 5.04404883892028181652436424424, 5.06154639951441437271598427158, 5.19936486244070317479880685098, 5.21539557044887741030676411961, 5.66146745069945851023919856596, 5.81097568073932704657902150934, 5.89950680602723040559802874398, 5.90986479284081894543370576759, 6.14930378360452239016544511183

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