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Properties

Label 8-1334e4-1.1-c1e4-0-1
Degree $8$
Conductor $3.167\times 10^{12}$
Sign $1$
Analytic cond. $12874.5$
Root an. cond. $3.26374$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 4·3-s + 10·4-s − 16·6-s − 4·7-s + 20·8-s + 2·9-s − 4·11-s − 40·12-s − 8·13-s − 16·14-s + 35·16-s − 12·17-s + 8·18-s − 16·19-s + 16·21-s − 16·22-s + 4·23-s − 80·24-s − 6·25-s − 32·26-s + 16·27-s − 40·28-s − 4·29-s − 12·31-s + 56·32-s + 16·33-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s − 6.53·6-s − 1.51·7-s + 7.07·8-s + 2/3·9-s − 1.20·11-s − 11.5·12-s − 2.21·13-s − 4.27·14-s + 35/4·16-s − 2.91·17-s + 1.88·18-s − 3.67·19-s + 3.49·21-s − 3.41·22-s + 0.834·23-s − 16.3·24-s − 6/5·25-s − 6.27·26-s + 3.07·27-s − 7.55·28-s − 0.742·29-s − 2.15·31-s + 9.89·32-s + 2.78·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 23^{4} \cdot 29^{4}\)
Sign: $1$
Analytic conductor: \(12874.5\)
Root analytic conductor: \(3.26374\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 23^{4} \cdot 29^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{4} \)
23$C_1$ \( ( 1 - T )^{4} \)
29$C_1$ \( ( 1 + T )^{4} \)
good3$D_{4}$ \( ( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 6 T^{2} - 16 T^{3} + 11 T^{4} - 16 p T^{5} + 6 p^{2} T^{6} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 26 T^{2} + 80 T^{3} + 268 T^{4} + 80 p T^{5} + 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 36 T^{2} + 120 T^{3} + 551 T^{4} + 120 p T^{5} + 36 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 4 p T^{2} + 216 T^{3} + 863 T^{4} + 216 p T^{5} + 4 p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 114 T^{2} + 664 T^{3} + 3284 T^{4} + 664 p T^{5} + 114 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 8 p T^{2} + 1008 T^{3} + 5058 T^{4} + 1008 p T^{5} + 8 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 140 T^{2} + 984 T^{3} + 6559 T^{4} + 984 p T^{5} + 140 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 90 T^{2} + 392 T^{3} + 4244 T^{4} + 392 p T^{5} + 90 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 146 T^{2} + 928 T^{3} + 7412 T^{4} + 928 p T^{5} + 146 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 92 T^{2} - 712 T^{3} + 4191 T^{4} - 712 p T^{5} + 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 28 T + 452 T^{2} + 4920 T^{3} + 39239 T^{4} + 4920 p T^{5} + 452 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 262 T^{2} - 2480 T^{3} + 22027 T^{4} - 2480 p T^{5} + 262 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 96 T^{2} - 356 T^{3} + 8790 T^{4} - 356 p T^{5} + 96 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 154 T^{2} + 536 T^{3} + 13444 T^{4} + 536 p T^{5} + 154 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 70 T^{2} + 212 T^{3} + 4388 T^{4} + 212 p T^{5} + 70 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 156 T^{2} - 768 T^{3} + 10806 T^{4} - 768 p T^{5} + 156 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 358 T^{2} + 3460 T^{3} + 31148 T^{4} + 3460 p T^{5} + 358 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 12 T + 230 T^{2} - 2368 T^{3} + 25851 T^{4} - 2368 p T^{5} + 230 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 312 T^{2} + 2684 T^{3} + 37958 T^{4} + 2684 p T^{5} + 312 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 182 T^{2} - 1388 T^{3} + 15868 T^{4} - 1388 p T^{5} + 182 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 296 T^{2} - 780 T^{3} + 38358 T^{4} - 780 p T^{5} + 296 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.92431228401404024499414905552, −6.83199577190481892073527462931, −6.59111148282513384770543980515, −6.47707202090867196456170947763, −6.44682681657032738807766712660, −6.12153387314983760478548074890, −5.82758020268768426382976689947, −5.73548171889145359333014879298, −5.68455468315646047436700756959, −5.18547149338347379988993788918, −5.02468937462851204800514285019, −4.91954562048224095651166034682, −4.84114764201517010577861671288, −4.58757921889251636294135376938, −4.25105782450021140826985657673, −3.98458211882716761654527408239, −3.89345538412091548834274852479, −3.27993263623403936055720840036, −3.20410202377490209395405270263, −3.06083417629139340583072236360, −2.62685257265831172501045137190, −2.22021489681909615341332417645, −2.09059551672174916897093206384, −2.08633651144937717060062834190, −1.64940485770766089669038888661, 0, 0, 0, 0, 1.64940485770766089669038888661, 2.08633651144937717060062834190, 2.09059551672174916897093206384, 2.22021489681909615341332417645, 2.62685257265831172501045137190, 3.06083417629139340583072236360, 3.20410202377490209395405270263, 3.27993263623403936055720840036, 3.89345538412091548834274852479, 3.98458211882716761654527408239, 4.25105782450021140826985657673, 4.58757921889251636294135376938, 4.84114764201517010577861671288, 4.91954562048224095651166034682, 5.02468937462851204800514285019, 5.18547149338347379988993788918, 5.68455468315646047436700756959, 5.73548171889145359333014879298, 5.82758020268768426382976689947, 6.12153387314983760478548074890, 6.44682681657032738807766712660, 6.47707202090867196456170947763, 6.59111148282513384770543980515, 6.83199577190481892073527462931, 6.92431228401404024499414905552

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