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Properties

Label 8-35e8-1.1-c3e4-0-4
Degree $8$
Conductor $2.252\times 10^{12}$
Sign $1$
Analytic cond. $2.72903\times 10^{7}$
Root an. cond. $8.50160$
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 − 3·3-s + 10·4-s + 12·6-s − 17·8-s − 19·9-s + 100·11-s − 30·12-s + 44·13-s − 10·16-s − 53·17-s + 76·18-s + 29·19-s − 400·22-s − 295·23-s + 51·24-s − 176·26-s − 54·27-s + 129·29-s − 114·31-s + 212·32-s − 300·33-s + 212·34-s − 190·36-s − 403·37-s − 116·38-s − 132·39-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 5/4·4-s + 0.816·6-s − 0.751·8-s − 0.703·9-s + 2.74·11-s − 0.721·12-s + 0.938·13-s − 0.156·16-s − 0.756·17-s + 0.995·18-s + 0.350·19-s − 3.87·22-s − 2.67·23-s + 0.433·24-s − 1.32·26-s − 0.384·27-s + 0.826·29-s − 0.660·31-s + 1.17·32-s − 1.58·33-s + 1.06·34-s − 0.879·36-s − 1.79·37-s − 0.495·38-s − 0.541·39-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(5^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.72903\times 10^{7}\)
Root analytic conductor: \(8.50160\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 5^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

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)$
bad5 \( 1 \)
7 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + p^{2} T + 3 p T^{2} + T^{3} + 11 p T^{4} + p^{3} T^{5} + 3 p^{7} T^{6} + p^{11} T^{7} + p^{12} T^{8} \)
3$C_2 \wr S_4$ \( 1 + p T + 28 T^{2} + 65 p T^{3} + 1150 T^{4} + 65 p^{4} T^{5} + 28 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \)
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

−6.90577747017518638876188848603, −6.86549271268285509763267693644, −6.73699440912720794576766934268, −6.31611469324170910101306684115, −6.10266817647913224128306225781, −5.94199711432685700350682523443, −5.88900146734510807040959223837, −5.59882092376569047550683865782, −5.47377215237084821810903600770, −4.90469260113944826905754662155, −4.74744492993242245056402390935, −4.43289551386949779488541864361, −4.17715055740484143193984492793, −3.93398383104174661985325003099, −3.88066246383256787242973696272, −3.67396005452955038873843986799, −3.13985392336095749798502308397, −2.90755977607247918896845824918, −2.73129773264815658185034573426, −2.38923472622279067456399828313, −1.76817491752824106549789722053, −1.64558723118750688615709789919, −1.45064870382712560056630914185, −1.35088420252162443431386266723, −1.06167432870786508679043778622, 0, 0, 0, 0, 1.06167432870786508679043778622, 1.35088420252162443431386266723, 1.45064870382712560056630914185, 1.64558723118750688615709789919, 1.76817491752824106549789722053, 2.38923472622279067456399828313, 2.73129773264815658185034573426, 2.90755977607247918896845824918, 3.13985392336095749798502308397, 3.67396005452955038873843986799, 3.88066246383256787242973696272, 3.93398383104174661985325003099, 4.17715055740484143193984492793, 4.43289551386949779488541864361, 4.74744492993242245056402390935, 4.90469260113944826905754662155, 5.47377215237084821810903600770, 5.59882092376569047550683865782, 5.88900146734510807040959223837, 5.94199711432685700350682523443, 6.10266817647913224128306225781, 6.31611469324170910101306684115, 6.73699440912720794576766934268, 6.86549271268285509763267693644, 6.90577747017518638876188848603

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