Properties

Label 8-1232e4-1.1-c1e4-0-4
Degree $8$
Conductor $2.304\times 10^{12}$
Sign $1$
Analytic cond. $9365.93$
Root an. cond. $3.13649$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5·5-s + 4·7-s − 9-s − 4·11-s + 4·13-s − 5·15-s + 10·17-s − 6·19-s − 4·21-s + 3·23-s + 11·25-s − 2·27-s − 4·29-s − 31-s + 4·33-s + 20·35-s + 13·37-s − 4·39-s + 10·41-s + 8·43-s − 5·45-s + 6·47-s + 10·49-s − 10·51-s + 8·53-s − 20·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 2.23·5-s + 1.51·7-s − 1/3·9-s − 1.20·11-s + 1.10·13-s − 1.29·15-s + 2.42·17-s − 1.37·19-s − 0.872·21-s + 0.625·23-s + 11/5·25-s − 0.384·27-s − 0.742·29-s − 0.179·31-s + 0.696·33-s + 3.38·35-s + 2.13·37-s − 0.640·39-s + 1.56·41-s + 1.21·43-s − 0.745·45-s + 0.875·47-s + 10/7·49-s − 1.40·51-s + 1.09·53-s − 2.69·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(9365.93\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1232} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(7.934308835\)
\(L(\frac12)\) \(\approx\) \(7.934308835\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - T )^{4} \)
11$C_1$ \( ( 1 + T )^{4} \)
good3$C_2^3:S_4$ \( 1 + T + 2 T^{2} + 5 T^{3} + 2 T^{4} + 5 p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
5$C_2^3:S_4$ \( 1 - p T + 14 T^{2} - 39 T^{3} + 98 T^{4} - 39 p T^{5} + 14 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 4 T + 20 T^{2} - 76 T^{3} + 438 T^{4} - 76 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 10 T + 84 T^{2} - 478 T^{3} + 2246 T^{4} - 478 p T^{5} + 84 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr S_4$ \( 1 + 6 T + 28 T^{2} + 22 T^{3} - 106 T^{4} + 22 p T^{5} + 28 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
23$S_4\times C_2$ \( 1 - 3 T + 40 T^{2} - 255 T^{3} + 846 T^{4} - 255 p T^{5} + 40 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 4 T + 36 T^{2} + 76 T^{3} + 710 T^{4} + 76 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr S_4$ \( 1 + T + 66 T^{2} + 65 T^{3} + 2322 T^{4} + 65 p T^{5} + 66 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 13 T + 170 T^{2} - 1343 T^{3} + 9898 T^{4} - 1343 p T^{5} + 170 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 10 T + 180 T^{2} - 1198 T^{3} + 11366 T^{4} - 1198 p T^{5} + 180 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 8 T + 76 T^{2} - 520 T^{3} + 4630 T^{4} - 520 p T^{5} + 76 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 6 T + 80 T^{2} - 10 p T^{3} + 2846 T^{4} - 10 p^{2} T^{5} + 80 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{4} \)
59$C_2 \wr S_4$ \( 1 + 3 T + 110 T^{2} + 439 T^{3} + 9050 T^{4} + 439 p T^{5} + 110 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 2 T + 180 T^{2} - 94 T^{3} + 14294 T^{4} - 94 p T^{5} + 180 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 3 T + 84 T^{2} - 549 T^{3} + 2406 T^{4} - 549 p T^{5} + 84 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 7 T + 192 T^{2} + 1667 T^{3} + 17118 T^{4} + 1667 p T^{5} + 192 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 2 T + 124 T^{2} - 6 p T^{3} + 12854 T^{4} - 6 p^{2} T^{5} + 124 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr S_4$ \( 1 - 46 T + 1068 T^{2} - 16022 T^{3} + 168294 T^{4} - 16022 p T^{5} + 1068 p^{2} T^{6} - 46 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 32 T + 636 T^{2} + 8448 T^{3} + 88214 T^{4} + 8448 p T^{5} + 636 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 7 T + 270 T^{2} - 1025 T^{3} + 30402 T^{4} - 1025 p T^{5} + 270 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 11 T + 246 T^{2} + 1381 T^{3} + 26034 T^{4} + 1381 p T^{5} + 246 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.99683072211486021620881482538, −6.53889250359933506748787050776, −6.34521508238139514503864389681, −6.06071013706725071166497866331, −5.99102058219094153914492796225, −5.73244155686669444474754164274, −5.68347886257387205332059830472, −5.54175409268434331586232891832, −5.38108489283253387841931016612, −4.99676871969677613410333814767, −4.68690061793333777283758403341, −4.57282762126806853540888097998, −4.34312595872253850564409591087, −4.04200732290514178253724867978, −3.55424762363695827510021215819, −3.45035890038223881888421563871, −3.19190842711923635156353232486, −2.66242507099111822202505792566, −2.40324190598383487418593773144, −2.21915610442674431129214057631, −2.10907772090319249006380140158, −1.67749676323175933775457421659, −1.20123658530616278012163953022, −0.946849228306040621616066085848, −0.67867894030819483583917532958, 0.67867894030819483583917532958, 0.946849228306040621616066085848, 1.20123658530616278012163953022, 1.67749676323175933775457421659, 2.10907772090319249006380140158, 2.21915610442674431129214057631, 2.40324190598383487418593773144, 2.66242507099111822202505792566, 3.19190842711923635156353232486, 3.45035890038223881888421563871, 3.55424762363695827510021215819, 4.04200732290514178253724867978, 4.34312595872253850564409591087, 4.57282762126806853540888097998, 4.68690061793333777283758403341, 4.99676871969677613410333814767, 5.38108489283253387841931016612, 5.54175409268434331586232891832, 5.68347886257387205332059830472, 5.73244155686669444474754164274, 5.99102058219094153914492796225, 6.06071013706725071166497866331, 6.34521508238139514503864389681, 6.53889250359933506748787050776, 6.99683072211486021620881482538

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