Properties

Label 8-7728e4-1.1-c1e4-0-2
Degree $8$
Conductor $3.567\times 10^{15}$
Sign $1$
Analytic cond. $1.45002\times 10^{7}$
Root an. cond. $7.85546$
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  + 4·3-s + 5·5-s + 4·7-s + 10·9-s − 11-s + 7·13-s + 20·15-s + 2·17-s + 19-s + 16·21-s − 4·23-s + 8·25-s + 20·27-s + 2·29-s − 6·31-s − 4·33-s + 20·35-s + 16·37-s + 28·39-s + 5·41-s − 9·43-s + 50·45-s + 21·47-s + 10·49-s + 8·51-s + 10·53-s − 5·55-s + ⋯
L(s)  = 1  + 2.30·3-s + 2.23·5-s + 1.51·7-s + 10/3·9-s − 0.301·11-s + 1.94·13-s + 5.16·15-s + 0.485·17-s + 0.229·19-s + 3.49·21-s − 0.834·23-s + 8/5·25-s + 3.84·27-s + 0.371·29-s − 1.07·31-s − 0.696·33-s + 3.38·35-s + 2.63·37-s + 4.48·39-s + 0.780·41-s − 1.37·43-s + 7.45·45-s + 3.06·47-s + 10/7·49-s + 1.12·51-s + 1.37·53-s − 0.674·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(1.45002\times 10^{7}\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7728} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(102.3553582\)
\(L(\frac12)\) \(\approx\) \(102.3553582\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
7$C_1$ \( ( 1 - T )^{4} \)
23$C_1$ \( ( 1 + T )^{4} \)
good5$C_2 \wr C_2\wr C_2$ \( 1 - p T + 17 T^{2} - 37 T^{3} + 88 T^{4} - 37 p T^{5} + 17 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + T + 27 T^{2} + 52 T^{3} + 356 T^{4} + 52 p T^{5} + 27 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 35 T^{2} - 109 T^{3} + 384 T^{4} - 109 p T^{5} + 35 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 39 T^{2} - 4 T^{3} + 684 T^{4} - 4 p T^{5} + 39 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - T + 59 T^{2} - 4 p T^{3} + 1524 T^{4} - 4 p^{2} T^{5} + 59 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
29$D_{4}$ \( ( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 6 T + 107 T^{2} + 548 T^{3} + 4728 T^{4} + 548 p T^{5} + 107 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 203 T^{2} - 1704 T^{3} + 12096 T^{4} - 1704 p T^{5} + 203 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 93 T^{2} - 798 T^{3} + 4130 T^{4} - 798 p T^{5} + 93 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 9 T + 169 T^{2} + 1033 T^{3} + 10820 T^{4} + 1033 p T^{5} + 169 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 21 T + 290 T^{2} - 2749 T^{3} + 21146 T^{4} - 2749 p T^{5} + 290 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 108 T^{2} - 1013 T^{3} + 9408 T^{4} - 1013 p T^{5} + 108 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 26 T + 484 T^{2} - 5629 T^{3} + 51706 T^{4} - 5629 p T^{5} + 484 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 64 T^{2} + 563 T^{3} + 1480 T^{4} + 563 p T^{5} + 64 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 5 T - 13 T^{2} - 391 T^{3} + 288 T^{4} - 391 p T^{5} - 13 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 19 T + 367 T^{2} - 4163 T^{3} + 42064 T^{4} - 4163 p T^{5} + 367 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 - 10 T + 231 T^{2} - 1964 T^{3} + 23676 T^{4} - 1964 p T^{5} + 231 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 6 T + 95 T^{2} - 1072 T^{3} + 11520 T^{4} - 1072 p T^{5} + 95 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 135 T^{2} - 368 T^{3} + 16784 T^{4} - 368 p T^{5} + 135 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 17 T + 181 T^{2} + 2227 T^{3} + 27564 T^{4} + 2227 p T^{5} + 181 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 32 T + 731 T^{2} - 10704 T^{3} + 124680 T^{4} - 10704 p T^{5} + 731 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \)
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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

−5.58955568251429811498983489535, −5.24519347285259693068286983220, −5.23579777329805449339148490397, −5.19501825936628098460691835598, −4.70684107124115494409635396712, −4.39317645585282890941837901331, −4.26302480931936082470429669681, −4.21437178076774970850713983530, −4.09038538584592480744241090129, −3.66730676847375082168734886218, −3.59217100121006549285649791427, −3.55818599900669183758151734857, −3.17974809303916313369342021820, −2.84422429082652816608461463835, −2.77608636978176645962163344564, −2.41757717377684280771078897433, −2.31790573184105091463959102782, −2.09426787200334321054346579232, −1.98849833132928814604858362612, −1.78647150633728923122749495521, −1.75571453526589894719714173959, −1.06510887790488622720664097977, −1.04912096756672502406369321281, −0.801699269782003185872189277361, −0.73633593168246341386534192271, 0.73633593168246341386534192271, 0.801699269782003185872189277361, 1.04912096756672502406369321281, 1.06510887790488622720664097977, 1.75571453526589894719714173959, 1.78647150633728923122749495521, 1.98849833132928814604858362612, 2.09426787200334321054346579232, 2.31790573184105091463959102782, 2.41757717377684280771078897433, 2.77608636978176645962163344564, 2.84422429082652816608461463835, 3.17974809303916313369342021820, 3.55818599900669183758151734857, 3.59217100121006549285649791427, 3.66730676847375082168734886218, 4.09038538584592480744241090129, 4.21437178076774970850713983530, 4.26302480931936082470429669681, 4.39317645585282890941837901331, 4.70684107124115494409635396712, 5.19501825936628098460691835598, 5.23579777329805449339148490397, 5.24519347285259693068286983220, 5.58955568251429811498983489535

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