Properties

Label 8-162e4-1.1-c8e4-0-2
Degree $8$
Conductor $688747536$
Sign $1$
Analytic cond. $1.89693\times 10^{7}$
Root an. cond. $8.12375$
Motivic weight $8$
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  + 128·4-s + 4.13e3·7-s − 1.61e4·13-s − 9.06e5·19-s − 3.20e5·25-s + 5.28e5·28-s − 1.65e6·31-s + 5.37e6·37-s + 1.22e7·43-s + 1.57e7·49-s − 2.06e6·52-s + 2.99e7·61-s − 2.09e6·64-s + 2.00e7·67-s − 9.30e7·73-s − 1.16e8·76-s − 2.85e7·79-s − 6.66e7·91-s + 8.11e7·97-s − 4.10e7·100-s + 2.74e8·103-s − 1.71e8·109-s − 3.84e8·121-s − 2.11e8·124-s + 127-s + 131-s − 3.74e9·133-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.72·7-s − 0.564·13-s − 6.95·19-s − 0.820·25-s + 0.860·28-s − 1.78·31-s + 2.86·37-s + 3.59·43-s + 2.73·49-s − 0.282·52-s + 2.16·61-s − 1/8·64-s + 0.994·67-s − 3.27·73-s − 3.47·76-s − 0.732·79-s − 0.971·91-s + 0.916·97-s − 0.410·100-s + 2.43·103-s − 1.21·109-s − 1.79·121-s − 0.894·124-s − 11.9·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(1.89693\times 10^{7}\)
Root analytic conductor: \(8.12375\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 4, 4, 4, 4 ),\ 1 )\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.3606874116\)
\(L(\frac12)\) \(\approx\) \(0.3606874116\)
\(L(5)\) 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$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
3 \( 1 \)
good5$C_2^3$ \( 1 + 12818 p^{2} T^{2} - 79839501 p^{4} T^{4} + 12818 p^{18} T^{6} + p^{32} T^{8} \)
7$C_2^2$ \( ( 1 - 295 p T - 30624 p^{2} T^{2} - 295 p^{9} T^{3} + p^{16} T^{4} )^{2} \)
11$C_2^3$ \( 1 + 384462530 T^{2} + 101861707110428739 T^{4} + 384462530 p^{16} T^{6} + p^{32} T^{8} \)
13$C_2^2$ \( ( 1 + 8063 T - 750718752 T^{2} + 8063 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
17$C_2^2$ \( ( 1 - 13485535490 T^{2} + p^{16} T^{4} )^{2} \)
19$C_2$ \( ( 1 + 226609 T + p^{8} T^{2} )^{4} \)
23$C_2^3$ \( 1 + 20955649154 T^{2} - \)\(56\!\cdots\!45\)\( T^{4} + 20955649154 p^{16} T^{6} + p^{32} T^{8} \)
29$C_2^3$ \( 1 + 145584434 p^{2} T^{2} - 332619955782368685 p^{4} T^{4} + 145584434 p^{18} T^{6} + p^{32} T^{8} \)
31$C_2^2$ \( ( 1 + 826370 T - 170003660541 T^{2} + 826370 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
37$C_2$ \( ( 1 - 1344575 T + p^{8} T^{2} )^{4} \)
41$C_2^3$ \( 1 - 10986038337406 T^{2} + \)\(56\!\cdots\!95\)\( T^{4} - 10986038337406 p^{16} T^{6} + p^{32} T^{8} \)
43$C_2^2$ \( ( 1 - 6147742 T + 26106531420963 T^{2} - 6147742 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
47$C_2^3$ \( 1 + 12685263482114 T^{2} - \)\(40\!\cdots\!25\)\( T^{4} + 12685263482114 p^{16} T^{6} + p^{32} T^{8} \)
53$C_2^2$ \( ( 1 - 123929317870274 T^{2} + p^{16} T^{4} )^{2} \)
59$C_2^3$ \( 1 + 293436242931650 T^{2} + \)\(64\!\cdots\!59\)\( T^{4} + 293436242931650 p^{16} T^{6} + p^{32} T^{8} \)
61$C_2^2$ \( ( 1 - 14985697 T + 32863801578528 T^{2} - 14985697 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 - 10023697 T - 305593176008832 T^{2} - 10023697 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 777369936279166 T^{2} + p^{16} T^{4} )^{2} \)
73$C_2$ \( ( 1 + 23261569 T + p^{8} T^{2} )^{4} \)
79$C_2^2$ \( ( 1 + 14267183 T - 1313556299151072 T^{2} + 14267183 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
83$C_2^3$ \( 1 + 3194735948169794 T^{2} + \)\(51\!\cdots\!55\)\( T^{4} + 3194735948169794 p^{16} T^{6} + p^{32} T^{8} \)
89$C_2^2$ \( ( 1 + 5372002023507070 T^{2} + p^{16} T^{4} )^{2} \)
97$C_2^2$ \( ( 1 - 40571617 T - 6191377488382272 T^{2} - 40571617 p^{8} T^{3} + p^{16} T^{4} )^{2} \)
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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

−7.87909665427348746062365363317, −7.74282039478722836401356770866, −7.27199244194900761217010138696, −7.16982791364752463820461031463, −6.57118734482855159547179272594, −6.57013352557210763496234707704, −5.96780053824130676085067126103, −5.94711591639743232410124412342, −5.91289443831002811182082318128, −5.30951718885098975359279722075, −4.76327154271673504564506459390, −4.59759683044826751584130098428, −4.38391983761612480431200232109, −4.01096097482061317383565361089, −3.88097846291488890780976102685, −3.83507975133273147629765612360, −2.52565491342958961405348416554, −2.45960995075670464769159767064, −2.35410875832564274099285127082, −2.29789121097122625720218751122, −1.80472632153134542664090237147, −1.34386033867077792913429051240, −1.08121458938280442915119743889, −0.43336802505935176518402528243, −0.084878001971269745288476880385, 0.084878001971269745288476880385, 0.43336802505935176518402528243, 1.08121458938280442915119743889, 1.34386033867077792913429051240, 1.80472632153134542664090237147, 2.29789121097122625720218751122, 2.35410875832564274099285127082, 2.45960995075670464769159767064, 2.52565491342958961405348416554, 3.83507975133273147629765612360, 3.88097846291488890780976102685, 4.01096097482061317383565361089, 4.38391983761612480431200232109, 4.59759683044826751584130098428, 4.76327154271673504564506459390, 5.30951718885098975359279722075, 5.91289443831002811182082318128, 5.94711591639743232410124412342, 5.96780053824130676085067126103, 6.57013352557210763496234707704, 6.57118734482855159547179272594, 7.16982791364752463820461031463, 7.27199244194900761217010138696, 7.74282039478722836401356770866, 7.87909665427348746062365363317

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