Properties

Label 8-162e4-1.1-c6e4-0-1
Degree $8$
Conductor $688747536$
Sign $1$
Analytic cond. $1.92921\times 10^{6}$
Root an. cond. $6.10481$
Motivic weight $6$
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  − 64·4-s − 4·7-s + 3.83e3·13-s + 3.07e3·16-s + 1.02e4·19-s + 1.84e4·25-s + 256·28-s − 4.00e4·31-s − 2.02e4·37-s + 1.84e5·43-s − 4.70e5·49-s − 2.45e5·52-s + 6.09e5·61-s − 1.31e5·64-s − 2.00e6·67-s + 5.25e5·73-s − 6.55e5·76-s − 8.48e5·79-s − 1.53e4·91-s + 3.62e6·97-s − 1.17e6·100-s + 5.27e6·103-s − 1.24e6·109-s − 1.22e4·112-s + 1.95e6·121-s + 2.56e6·124-s + 127-s + ⋯
L(s)  = 1  − 4-s − 0.0116·7-s + 1.74·13-s + 3/4·16-s + 1.49·19-s + 1.17·25-s + 0.0116·28-s − 1.34·31-s − 0.398·37-s + 2.32·43-s − 3.99·49-s − 1.74·52-s + 2.68·61-s − 1/2·64-s − 6.67·67-s + 1.35·73-s − 1.49·76-s − 1.72·79-s − 0.0203·91-s + 3.96·97-s − 1.17·100-s + 4.82·103-s − 0.959·109-s − 0.00874·112-s + 1.10·121-s + 1.34·124-s − 0.0174·133-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(1.92921\times 10^{6}\)
Root analytic conductor: \(6.10481\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{16} ,\ ( \ : 3, 3, 3, 3 ),\ 1 )\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(4.389141112\)
\(L(\frac12)\) \(\approx\) \(4.389141112\)
\(L(4)\) 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$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
3 \( 1 \)
good5$D_4\times C_2$ \( 1 - 736 p^{2} T^{2} + 181599 p^{4} T^{4} - 736 p^{14} T^{6} + p^{24} T^{8} \)
7$D_{4}$ \( ( 1 + 2 T + 235272 T^{2} + 2 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
11$D_4\times C_2$ \( 1 - 1952680 T^{2} + 729391264242 T^{4} - 1952680 p^{12} T^{6} + p^{24} T^{8} \)
13$D_{4}$ \( ( 1 - 1918 T + 7035111 T^{2} - 1918 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 81233464 T^{2} + 2800570401456999 T^{4} - 81233464 p^{12} T^{6} + p^{24} T^{8} \)
19$D_{4}$ \( ( 1 - 5122 T + 92571840 T^{2} - 5122 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 306194656 T^{2} + 48390197876780226 T^{4} - 306194656 p^{12} T^{6} + p^{24} T^{8} \)
29$D_4\times C_2$ \( 1 + 218801912 T^{2} + 713450102444654343 T^{4} + 218801912 p^{12} T^{6} + p^{24} T^{8} \)
31$D_{4}$ \( ( 1 + 20048 T + 1659858486 T^{2} + 20048 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 10100 T + 5013752475 T^{2} + 10100 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 2589640888 T^{2} + 18656732024545255890 T^{4} - 2589640888 p^{12} T^{6} + p^{24} T^{8} \)
43$D_{4}$ \( ( 1 - 92470 T + 8867120496 T^{2} - 92470 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 27858666964 T^{2} + 191878190860258806 p^{2} T^{4} - 27858666964 p^{12} T^{6} + p^{24} T^{8} \)
53$D_4\times C_2$ \( 1 - 51925621720 T^{2} + \)\(16\!\cdots\!94\)\( T^{4} - 51925621720 p^{12} T^{6} + p^{24} T^{8} \)
59$D_4\times C_2$ \( 1 - 144480622612 T^{2} + \)\(86\!\cdots\!26\)\( T^{4} - 144480622612 p^{12} T^{6} + p^{24} T^{8} \)
61$D_{4}$ \( ( 1 - 304528 T + 97865805831 T^{2} - 304528 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 + 1004486 T + 432404360664 T^{2} + 1004486 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 94765336960 T^{2} + \)\(38\!\cdots\!74\)\( T^{4} - 94765336960 p^{12} T^{6} + p^{24} T^{8} \)
73$D_{4}$ \( ( 1 - 262912 T + 52515591 p^{2} T^{2} - 262912 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 424358 T + 63430257600 T^{2} + 424358 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 413441187988 T^{2} + \)\(57\!\cdots\!86\)\( T^{4} - 413441187988 p^{12} T^{6} + p^{24} T^{8} \)
89$D_4\times C_2$ \( 1 - 332250007216 T^{2} + \)\(48\!\cdots\!59\)\( T^{4} - 332250007216 p^{12} T^{6} + p^{24} T^{8} \)
97$D_{4}$ \( ( 1 - 1810864 T + 2248345168182 T^{2} - 1810864 p^{6} T^{3} + p^{12} T^{4} )^{2} \)
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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

−8.470500397925486215641505829311, −7.965692940750314557097277917638, −7.69911966681077427095742372652, −7.40765341690456207880144272247, −7.38360132305516723843919135754, −6.83792085026262081554111260614, −6.58877983235400973149546385791, −6.04789163722634940481002238358, −6.01337324632194373119299402676, −5.67136035591161315748859067512, −5.51357133450215452671542029177, −4.87354585465978095580975210731, −4.70088089819609813924858913189, −4.55643415361811843409152634271, −4.14925118757044234608576974113, −3.55102942194812949667365946150, −3.41748306930860739481483671933, −3.15585639334753910017239072969, −2.97299123776464371834841582617, −2.15065574576836204074426493832, −1.75135500212716955847425025014, −1.44217208066409831776983601739, −1.02765213242165244982656942075, −0.60825224092598202943167468638, −0.38030557152875619794465872017, 0.38030557152875619794465872017, 0.60825224092598202943167468638, 1.02765213242165244982656942075, 1.44217208066409831776983601739, 1.75135500212716955847425025014, 2.15065574576836204074426493832, 2.97299123776464371834841582617, 3.15585639334753910017239072969, 3.41748306930860739481483671933, 3.55102942194812949667365946150, 4.14925118757044234608576974113, 4.55643415361811843409152634271, 4.70088089819609813924858913189, 4.87354585465978095580975210731, 5.51357133450215452671542029177, 5.67136035591161315748859067512, 6.01337324632194373119299402676, 6.04789163722634940481002238358, 6.58877983235400973149546385791, 6.83792085026262081554111260614, 7.38360132305516723843919135754, 7.40765341690456207880144272247, 7.69911966681077427095742372652, 7.965692940750314557097277917638, 8.470500397925486215641505829311

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