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Properties

Label 8-75e8-1.1-c1e4-0-3
Degree $8$
Conductor $1.001\times 10^{15}$
Sign $1$
Analytic cond. $4.07003\times 10^{6}$
Root an. cond. $6.70192$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 2·7-s + 5·8-s + 7·11-s − 13-s − 4·14-s + 5·16-s + 2·17-s + 5·19-s + 14·22-s + 23-s − 2·26-s − 4·28-s + 20·29-s + 23·31-s − 2·32-s + 4·34-s − 2·37-s + 10·38-s + 12·41-s − 16·43-s + 14·44-s + 2·46-s + 2·47-s − 8·49-s − 2·52-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.755·7-s + 1.76·8-s + 2.11·11-s − 0.277·13-s − 1.06·14-s + 5/4·16-s + 0.485·17-s + 1.14·19-s + 2.98·22-s + 0.208·23-s − 0.392·26-s − 0.755·28-s + 3.71·29-s + 4.13·31-s − 0.353·32-s + 0.685·34-s − 0.328·37-s + 1.62·38-s + 1.87·41-s − 2.43·43-s + 2.11·44-s + 0.294·46-s + 0.291·47-s − 8/7·49-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 5^{16}\)
Sign: $1$
Analytic conductor: \(4.07003\times 10^{6}\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(35.44304599\)
\(L(\frac12)\) \(\approx\) \(35.44304599\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good2$C_2^2:C_4$ \( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \)
7$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 12 T^{2} + 15 T^{3} + 61 T^{4} + 15 p T^{5} + 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
11$C_2 \wr C_2\wr C_2$ \( 1 - 7 T + 38 T^{2} - 139 T^{3} + 485 T^{4} - 139 p T^{5} + 38 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2 \wr C_2\wr C_2$ \( 1 + T + 38 T^{2} + 15 T^{3} + 641 T^{4} + 15 p T^{5} + 38 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
17$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 62 T^{2} - 95 T^{3} + 1541 T^{4} - 95 p T^{5} + 62 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_2 \wr C_2\wr C_2$ \( 1 - 5 T + 41 T^{2} - 210 T^{3} + 1111 T^{4} - 210 p T^{5} + 41 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \)
23$C_2 \wr C_2\wr C_2$ \( 1 - T + 68 T^{2} - 5 p T^{3} + 2051 T^{4} - 5 p^{2} T^{5} + 68 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \)
29$C_2 \wr C_2\wr C_2$ \( 1 - 20 T + 256 T^{2} - 2145 T^{3} + 13571 T^{4} - 2145 p T^{5} + 256 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \)
31$C_2 \wr C_2\wr C_2$ \( 1 - 23 T + 308 T^{2} - 2751 T^{3} + 17885 T^{4} - 2751 p T^{5} + 308 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 102 T^{2} + 175 T^{3} + 4811 T^{4} + 175 p T^{5} + 102 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
41$D_{4}$ \( ( 1 - 6 T + 71 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \)
43$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 233 T^{2} + 2040 T^{3} + 16241 T^{4} + 2040 p T^{5} + 233 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
47$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 152 T^{2} - 245 T^{3} + 10181 T^{4} - 245 p T^{5} + 152 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2 \wr C_2\wr C_2$ \( 1 + 4 T + 178 T^{2} + 585 T^{3} + 13511 T^{4} + 585 p T^{5} + 178 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \)
59$C_2 \wr C_2\wr C_2$ \( 1 - 15 T + 191 T^{2} - 1440 T^{3} + 11931 T^{4} - 1440 p T^{5} + 191 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
61$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 188 T^{2} + 444 T^{3} + 15485 T^{4} + 444 p T^{5} + 188 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
67$C_2 \wr C_2\wr C_2$ \( 1 + 2 T + 237 T^{2} + 390 T^{3} + 22951 T^{4} + 390 p T^{5} + 237 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
71$C_2 \wr C_2\wr C_2$ \( 1 - 2 T + 68 T^{2} + 811 T^{3} + 485 T^{4} + 811 p T^{5} + 68 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2 \wr C_2\wr C_2$ \( 1 + 16 T + 243 T^{2} + 2010 T^{3} + 19951 T^{4} + 2010 p T^{5} + 243 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \)
79$C_2 \wr C_2\wr C_2$ \( 1 - 35 T + 636 T^{2} - 7945 T^{3} + 78161 T^{4} - 7945 p T^{5} + 636 p^{2} T^{6} - 35 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2 \wr C_2\wr C_2$ \( 1 - 16 T + 278 T^{2} - 2680 T^{3} + 31391 T^{4} - 2680 p T^{5} + 278 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \)
89$C_2 \wr C_2\wr C_2$ \( 1 - 35 T + 776 T^{2} - 11375 T^{3} + 125591 T^{4} - 11375 p T^{5} + 776 p^{2} T^{6} - 35 p^{3} T^{7} + p^{4} T^{8} \)
97$C_2 \wr C_2\wr C_2$ \( 1 + 12 T + 302 T^{2} + 2760 T^{3} + 41871 T^{4} + 2760 p T^{5} + 302 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \)
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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

−6.00448208338784123385406479308, −5.22891045781653922806460145256, −5.12978122329289916083181586805, −5.10912005741535763079117696596, −4.96906111261519119807306519049, −4.63167097784534082424624271102, −4.61438942208781553252984715610, −4.52077000952117757851238705453, −4.39217451081510228781874212782, −3.81670956272429660372079252549, −3.71419072338820748701320828026, −3.68785270658764660678548115392, −3.57365178332189312711477260012, −3.14656785482883504919021603744, −3.00888240883056964778662068318, −2.80911642572304751321719322983, −2.46880329714917171912261187993, −2.36095498268214426723343650657, −2.24368132036481642322173446927, −1.58116148754028389986090257311, −1.55502763270530494340333547851, −1.24727451567804795985022590611, −0.842435650931887921172944530307, −0.796607742887224104024380111421, −0.58683403211265121354573265555, 0.58683403211265121354573265555, 0.796607742887224104024380111421, 0.842435650931887921172944530307, 1.24727451567804795985022590611, 1.55502763270530494340333547851, 1.58116148754028389986090257311, 2.24368132036481642322173446927, 2.36095498268214426723343650657, 2.46880329714917171912261187993, 2.80911642572304751321719322983, 3.00888240883056964778662068318, 3.14656785482883504919021603744, 3.57365178332189312711477260012, 3.68785270658764660678548115392, 3.71419072338820748701320828026, 3.81670956272429660372079252549, 4.39217451081510228781874212782, 4.52077000952117757851238705453, 4.61438942208781553252984715610, 4.63167097784534082424624271102, 4.96906111261519119807306519049, 5.10912005741535763079117696596, 5.12978122329289916083181586805, 5.22891045781653922806460145256, 6.00448208338784123385406479308

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