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Properties

Label 8-1575e4-1.1-c3e4-0-4
Degree $8$
Conductor $6.154\times 10^{12}$
Sign $1$
Analytic cond. $7.45738\times 10^{7}$
Root an. cond. $9.63991$
Motivic weight $3$
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  + 4·2-s + 10·4-s − 28·7-s + 17·8-s − 100·11-s − 44·13-s − 112·14-s − 10·16-s − 53·17-s − 29·19-s − 400·22-s + 295·23-s − 176·26-s − 280·28-s − 129·29-s + 114·31-s − 212·32-s − 212·34-s − 403·37-s − 116·38-s − 671·41-s + 411·43-s − 1.00e3·44-s + 1.18e3·46-s − 8·47-s + 490·49-s − 440·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 5/4·4-s − 1.51·7-s + 0.751·8-s − 2.74·11-s − 0.938·13-s − 2.13·14-s − 0.156·16-s − 0.756·17-s − 0.350·19-s − 3.87·22-s + 2.67·23-s − 1.32·26-s − 1.88·28-s − 0.826·29-s + 0.660·31-s − 1.17·32-s − 1.06·34-s − 1.79·37-s − 0.495·38-s − 2.55·41-s + 1.45·43-s − 3.42·44-s + 3.78·46-s − 0.0248·47-s + 10/7·49-s − 1.17·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(7.45738\times 10^{7}\)
Root analytic conductor: \(9.63991\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 3^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + p T )^{4} \)
good2$C_2 \wr S_4$ \( 1 - p^{2} T + 3 p T^{2} - T^{3} + 11 p T^{4} - p^{3} T^{5} + 3 p^{7} T^{6} - p^{11} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 100 T + 4738 T^{2} + 106848 T^{3} + 2242403 T^{4} + 106848 p^{3} T^{5} + 4738 p^{6} T^{6} + 100 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 + 44 T + 4820 T^{2} + 154316 T^{3} + 10485014 T^{4} + 154316 p^{3} T^{5} + 4820 p^{6} T^{6} + 44 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 53 T + 15496 T^{2} + 576511 T^{3} + 103902926 T^{4} + 576511 p^{3} T^{5} + 15496 p^{6} T^{6} + 53 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 29 T + 6638 T^{2} + 258553 T^{3} + 53805642 T^{4} + 258553 p^{3} T^{5} + 6638 p^{6} T^{6} + 29 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 295 T + 54367 T^{2} - 7704248 T^{3} + 887991584 T^{4} - 7704248 p^{3} T^{5} + 54367 p^{6} T^{6} - 295 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 129 T + 83933 T^{2} + 9167698 T^{3} + 2922882282 T^{4} + 9167698 p^{3} T^{5} + 83933 p^{6} T^{6} + 129 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 - 114 T + 80728 T^{2} - 5927082 T^{3} + 3079701134 T^{4} - 5927082 p^{3} T^{5} + 80728 p^{6} T^{6} - 114 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 + 403 T + 202319 T^{2} + 46625606 T^{3} + 14127493020 T^{4} + 46625606 p^{3} T^{5} + 202319 p^{6} T^{6} + 403 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 671 T + 399120 T^{2} + 139154877 T^{3} + 44432315734 T^{4} + 139154877 p^{3} T^{5} + 399120 p^{6} T^{6} + 671 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 411 T + 213439 T^{2} - 60368868 T^{3} + 23440158536 T^{4} - 60368868 p^{3} T^{5} + 213439 p^{6} T^{6} - 411 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 8 T + 160820 T^{2} - 20939512 T^{3} + 12301264614 T^{4} - 20939512 p^{3} T^{5} + 160820 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 90 T + 133752 T^{2} + 19514962 T^{3} + 19633648894 T^{4} + 19514962 p^{3} T^{5} + 133752 p^{6} T^{6} - 90 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 1018 T + 1189512 T^{2} + 683319466 T^{3} + 407279566174 T^{4} + 683319466 p^{3} T^{5} + 1189512 p^{6} T^{6} + 1018 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 50 T + 722188 T^{2} - 51306118 T^{3} + 225994176278 T^{4} - 51306118 p^{3} T^{5} + 722188 p^{6} T^{6} - 50 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 424 T + 917806 T^{2} + 255868016 T^{3} + 369880128047 T^{4} + 255868016 p^{3} T^{5} + 917806 p^{6} T^{6} + 424 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 215 T + 982021 T^{2} + 275350320 T^{3} + 449702043326 T^{4} + 275350320 p^{3} T^{5} + 982021 p^{6} T^{6} + 215 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1207 T + 1581740 T^{2} - 1193773973 T^{3} + 948033109470 T^{4} - 1193773973 p^{3} T^{5} + 1581740 p^{6} T^{6} - 1207 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 951 T + 1431057 T^{2} + 665911232 T^{3} + 746318767554 T^{4} + 665911232 p^{3} T^{5} + 1431057 p^{6} T^{6} + 951 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 3035 T + 5488190 T^{2} + 6563721455 T^{3} + 5816729995458 T^{4} + 6563721455 p^{3} T^{5} + 5488190 p^{6} T^{6} + 3035 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 2819 T + 5531118 T^{2} + 7041312813 T^{3} + 6955665197242 T^{4} + 7041312813 p^{3} T^{5} + 5531118 p^{6} T^{6} + 2819 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 - 1100 T + 625060 T^{2} + 1205286580 T^{3} - 1327506690522 T^{4} + 1205286580 p^{3} T^{5} + 625060 p^{6} T^{6} - 1100 p^{9} T^{7} + p^{12} T^{8} \)
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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

−6.83522105699567632849114607417, −6.66547057210302884916346173312, −6.08445837208276198508666334141, −6.07130404046609051719066019444, −5.87399555005394543990128888500, −5.70383852807925999194601332644, −5.27513364180265903058296641855, −5.21781657259045096229567898352, −5.21095000228852869295322055771, −4.66043961227287916805022355410, −4.63451614880122821168500082308, −4.42059033056585720743687983892, −4.39137653105078944536123181915, −3.77033592552118968377450768158, −3.43658528816878639627630275759, −3.42287669883448047211874415962, −3.06682260160857588527523355090, −2.93597061187632528117779620375, −2.73288405632729858763018071585, −2.62286434407456940370808181808, −2.17134385187422557534768265302, −1.85440271616855420699768383103, −1.83034871472492507952397496032, −1.03205557113987557879408677893, −1.01705184489022442378719325199, 0, 0, 0, 0, 1.01705184489022442378719325199, 1.03205557113987557879408677893, 1.83034871472492507952397496032, 1.85440271616855420699768383103, 2.17134385187422557534768265302, 2.62286434407456940370808181808, 2.73288405632729858763018071585, 2.93597061187632528117779620375, 3.06682260160857588527523355090, 3.42287669883448047211874415962, 3.43658528816878639627630275759, 3.77033592552118968377450768158, 4.39137653105078944536123181915, 4.42059033056585720743687983892, 4.63451614880122821168500082308, 4.66043961227287916805022355410, 5.21095000228852869295322055771, 5.21781657259045096229567898352, 5.27513364180265903058296641855, 5.70383852807925999194601332644, 5.87399555005394543990128888500, 6.07130404046609051719066019444, 6.08445837208276198508666334141, 6.66547057210302884916346173312, 6.83522105699567632849114607417

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