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Properties

Label 8-1666e4-1.1-c1e4-0-7
Degree $8$
Conductor $7.704\times 10^{12}$
Sign $1$
Analytic cond. $31319.0$
Root an. cond. $3.64733$
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 − 8·5-s − 16·6-s + 20·8-s + 6·9-s − 32·10-s − 4·11-s − 40·12-s − 8·13-s + 32·15-s + 35·16-s + 4·17-s + 24·18-s − 8·19-s − 80·20-s − 16·22-s + 4·23-s − 80·24-s + 26·25-s − 32·26-s − 4·29-s + 128·30-s − 4·31-s + 56·32-s + 16·33-s + ⋯
L(s)  = 1  + 2.82·2-s − 2.30·3-s + 5·4-s − 3.57·5-s − 6.53·6-s + 7.07·8-s + 2·9-s − 10.1·10-s − 1.20·11-s − 11.5·12-s − 2.21·13-s + 8.26·15-s + 35/4·16-s + 0.970·17-s + 5.65·18-s − 1.83·19-s − 17.8·20-s − 3.41·22-s + 0.834·23-s − 16.3·24-s + 26/5·25-s − 6.27·26-s − 0.742·29-s + 23.3·30-s − 0.718·31-s + 9.89·32-s + 2.78·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 7^{8} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(31319.0\)
Root analytic conductor: \(3.64733\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 7^{8} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

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$C_1$ \( ( 1 - T )^{4} \)
7 \( 1 \)
17$C_1$ \( ( 1 - T )^{4} \)
good3$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 10 T^{2} + 16 T^{3} + 28 T^{4} + 16 p T^{5} + 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
5$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 38 T^{2} + 124 T^{3} + 316 T^{4} + 124 p T^{5} + 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 34 T^{2} + 128 T^{3} + 508 T^{4} + 128 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + 402 T^{4} + 8 p^{2} T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 56 T^{2} + 184 T^{3} + 898 T^{4} + 184 p T^{5} + 56 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - 4 T + 72 T^{2} - 196 T^{3} + 2198 T^{4} - 196 p T^{5} + 72 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 42 T^{2} + 88 T^{3} + 740 T^{4} + 88 p T^{5} + 42 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 + 4 T - 16 T^{2} + 20 T^{3} + 1510 T^{4} + 20 p T^{5} - 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 350 T^{2} + 3372 T^{3} + 24028 T^{4} + 3372 p T^{5} + 350 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2 \wr C_2\wr C_2$ \( 1 + 20 T + 296 T^{2} + 2812 T^{3} + 518 p T^{4} + 2812 p T^{5} + 296 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 116 T^{2} - 96 T^{3} + 6310 T^{4} - 96 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 + 96 T^{2} - 256 T^{3} + 4802 T^{4} - 256 p T^{5} + 96 p^{2} T^{6} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 92 T^{2} + 32 T^{3} + 7238 T^{4} + 32 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 240 T^{2} + 2352 T^{3} + 20210 T^{4} + 2352 p T^{5} + 240 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 - 8 T + 142 T^{2} - 1164 T^{3} + 9884 T^{4} - 1164 p T^{5} + 142 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 140 T^{2} - 320 T^{3} + 10950 T^{4} - 320 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 + 132 T^{2} + 96 T^{3} + 10230 T^{4} + 96 p T^{5} + 132 p^{2} T^{6} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 24 T + 484 T^{2} + 5800 T^{3} + 60278 T^{4} + 5800 p T^{5} + 484 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 236 T^{2} + 1616 T^{3} + 16678 T^{4} + 1616 p T^{5} + 236 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 296 T^{2} + 1912 T^{3} + 35554 T^{4} + 1912 p T^{5} + 296 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 + 8 T + 112 T^{2} + 1288 T^{3} + 12898 T^{4} + 1288 p T^{5} + 112 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 64 T^{2} + 1340 T^{3} + 2950 T^{4} + 1340 p T^{5} + 64 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
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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

−7.06423456448576948472029104545, −7.02772627248661257091546609176, −6.48857577924769876844641354077, −6.37258709394814155075299957125, −6.26536329103331834805135182851, −5.85978115361817090588730374779, −5.49370272967643084015676539455, −5.47529917220800376672185100333, −5.29212675831128658435062119710, −5.07906121188548860611920190606, −5.04053023471124077601507069598, −4.69188386580679700898774801703, −4.67292839866223092867975001861, −4.10377001600164060349841719043, −4.08686178829014208630738701954, −4.02198003496441706942151364587, −3.75198395862321917845492658305, −3.33517907265392941391658786633, −3.18753451670116660337467194663, −2.93053183978779201141675142622, −2.83794548837627852683041993682, −2.35516776436715764181078929268, −1.96786227787310923256903771402, −1.46947377454469504894934529244, −1.42592714036262431727603830646, 0, 0, 0, 0, 1.42592714036262431727603830646, 1.46947377454469504894934529244, 1.96786227787310923256903771402, 2.35516776436715764181078929268, 2.83794548837627852683041993682, 2.93053183978779201141675142622, 3.18753451670116660337467194663, 3.33517907265392941391658786633, 3.75198395862321917845492658305, 4.02198003496441706942151364587, 4.08686178829014208630738701954, 4.10377001600164060349841719043, 4.67292839866223092867975001861, 4.69188386580679700898774801703, 5.04053023471124077601507069598, 5.07906121188548860611920190606, 5.29212675831128658435062119710, 5.47529917220800376672185100333, 5.49370272967643084015676539455, 5.85978115361817090588730374779, 6.26536329103331834805135182851, 6.37258709394814155075299957125, 6.48857577924769876844641354077, 7.02772627248661257091546609176, 7.06423456448576948472029104545

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