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Properties

Label 8-483e4-1.1-c1e4-0-1
Degree $8$
Conductor $54423757521$
Sign $1$
Analytic cond. $221.256$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·3-s + 5·5-s − 8·6-s − 4·7-s − 8-s + 10·9-s + 10·10-s + 11-s + 7·13-s − 8·14-s − 20·15-s + 2·16-s + 2·17-s + 20·18-s − 19-s + 16·21-s + 2·22-s + 4·23-s + 4·24-s + 8·25-s + 14·26-s − 20·27-s + 2·29-s − 40·30-s + 6·31-s + 2·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 2.30·3-s + 2.23·5-s − 3.26·6-s − 1.51·7-s − 0.353·8-s + 10/3·9-s + 3.16·10-s + 0.301·11-s + 1.94·13-s − 2.13·14-s − 5.16·15-s + 1/2·16-s + 0.485·17-s + 4.71·18-s − 0.229·19-s + 3.49·21-s + 0.426·22-s + 0.834·23-s + 0.816·24-s + 8/5·25-s + 2.74·26-s − 3.84·27-s + 0.371·29-s − 7.30·30-s + 1.07·31-s + 0.353·32-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 7^{4} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(221.256\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 7^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(3.312821655\)
\(L(\frac12)\) \(\approx\) \(3.312821655\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 + T )^{4} \)
7$C_1$ \( ( 1 + T )^{4} \)
23$C_1$ \( ( 1 - T )^{4} \)
good2$C_2 \wr C_2\wr C_2$ \( 1 - p T + p^{2} T^{2} - 7 T^{3} + 5 p T^{4} - 7 p T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 - p T + 17 T^{2} - 37 T^{3} + 88 T^{4} - 37 p T^{5} + 17 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
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

−7.68643466691953921231116998944, −7.64533888974531840341083698971, −7.37047365136649322702300170112, −6.91245582849953401717626643510, −6.51056154740179898475940086629, −6.33161498770041872464772859410, −6.30731291380189075104245450433, −6.30451106493942517353735950746, −5.95983080104361308225981473247, −5.70672545216908292895950509535, −5.41944127237424158004100042323, −5.32095843718071002629300042888, −5.12897949420093551423785093936, −4.39685572342532568633757793783, −4.33552640817428854407863321065, −4.31157475443596607822001544594, −4.29606734312514636847746370997, −3.24468222832909204187764363912, −3.22926501331519815345422729699, −3.22512993107701523218861468178, −2.36780115030343750314389151899, −2.01093032880331381040071917599, −1.47162342059259279481303342154, −1.07446734187228023635943245477, −0.70763662990958991157801470531, 0.70763662990958991157801470531, 1.07446734187228023635943245477, 1.47162342059259279481303342154, 2.01093032880331381040071917599, 2.36780115030343750314389151899, 3.22512993107701523218861468178, 3.22926501331519815345422729699, 3.24468222832909204187764363912, 4.29606734312514636847746370997, 4.31157475443596607822001544594, 4.33552640817428854407863321065, 4.39685572342532568633757793783, 5.12897949420093551423785093936, 5.32095843718071002629300042888, 5.41944127237424158004100042323, 5.70672545216908292895950509535, 5.95983080104361308225981473247, 6.30451106493942517353735950746, 6.30731291380189075104245450433, 6.33161498770041872464772859410, 6.51056154740179898475940086629, 6.91245582849953401717626643510, 7.37047365136649322702300170112, 7.64533888974531840341083698971, 7.68643466691953921231116998944

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