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Properties

Label 8-930e4-1.1-c3e4-0-3
Degree $8$
Conductor $748052010000$
Sign $1$
Analytic cond. $9.06559\times 10^{6}$
Root an. cond. $7.40754$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $4$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 12·3-s + 40·4-s − 20·5-s + 96·6-s + 14·7-s − 160·8-s + 90·9-s + 160·10-s − 96·11-s − 480·12-s + 72·13-s − 112·14-s + 240·15-s + 560·16-s − 48·17-s − 720·18-s + 76·19-s − 800·20-s − 168·21-s + 768·22-s + 52·23-s + 1.92e3·24-s + 250·25-s − 576·26-s − 540·27-s + 560·28-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s + 0.755·7-s − 7.07·8-s + 10/3·9-s + 5.05·10-s − 2.63·11-s − 11.5·12-s + 1.53·13-s − 2.13·14-s + 4.13·15-s + 35/4·16-s − 0.684·17-s − 9.42·18-s + 0.917·19-s − 8.94·20-s − 1.74·21-s + 7.44·22-s + 0.471·23-s + 16.3·24-s + 2·25-s − 4.34·26-s − 3.84·27-s + 3.77·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\)
Sign: $1$
Analytic conductor: \(9.06559\times 10^{6}\)
Root analytic conductor: \(7.40754\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(4\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{4} \)
3$C_1$ \( ( 1 + p T )^{4} \)
5$C_1$ \( ( 1 + p T )^{4} \)
31$C_1$ \( ( 1 - p T )^{4} \)
good7$C_2 \wr S_4$ \( 1 - 2 p T + 534 T^{2} - 456 T^{3} + 100274 T^{4} - 456 p^{3} T^{5} + 534 p^{6} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 96 T + 7108 T^{2} + 371264 T^{3} + 1391362 p T^{4} + 371264 p^{3} T^{5} + 7108 p^{6} T^{6} + 96 p^{9} T^{7} + p^{12} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 72 T + 9514 T^{2} - 454570 T^{3} + 32008006 T^{4} - 454570 p^{3} T^{5} + 9514 p^{6} T^{6} - 72 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 48 T + 11308 T^{2} + 769180 T^{3} + 68497910 T^{4} + 769180 p^{3} T^{5} + 11308 p^{6} T^{6} + 48 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 4 p T + 14620 T^{2} - 1021452 T^{3} + 157178998 T^{4} - 1021452 p^{3} T^{5} + 14620 p^{6} T^{6} - 4 p^{10} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 52 T + 30428 T^{2} - 2534432 T^{3} + 439027374 T^{4} - 2534432 p^{3} T^{5} + 30428 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 104 T + 76028 T^{2} + 248402 p T^{3} + 2530905150 T^{4} + 248402 p^{4} T^{5} + 76028 p^{6} T^{6} + 104 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 174 T + 145350 T^{2} - 22610156 T^{3} + 10391407862 T^{4} - 22610156 p^{3} T^{5} + 145350 p^{6} T^{6} - 174 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 242 T + 91364 T^{2} + 23818054 T^{3} + 6959384550 T^{4} + 23818054 p^{3} T^{5} + 91364 p^{6} T^{6} + 242 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 754 T + 9824 p T^{2} - 149658298 T^{3} + 48482988238 T^{4} - 149658298 p^{3} T^{5} + 9824 p^{7} T^{6} - 754 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 + 118 T + 35548 T^{2} + 13396402 T^{3} + 13970282390 T^{4} + 13396402 p^{3} T^{5} + 35548 p^{6} T^{6} + 118 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 526 T + 278664 T^{2} - 110793062 T^{3} + 31875918934 T^{4} - 110793062 p^{3} T^{5} + 278664 p^{6} T^{6} - 526 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 106 T + 569508 T^{2} + 111297212 T^{3} + 147272057110 T^{4} + 111297212 p^{3} T^{5} + 569508 p^{6} T^{6} + 106 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 - 338 T + 650880 T^{2} - 165680262 T^{3} + 199323683918 T^{4} - 165680262 p^{3} T^{5} + 650880 p^{6} T^{6} - 338 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 320 T + 674754 T^{2} - 290886162 T^{3} + 257456485106 T^{4} - 290886162 p^{3} T^{5} + 674754 p^{6} T^{6} - 320 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 628 T + 1147092 T^{2} + 640763774 T^{3} + 572749428550 T^{4} + 640763774 p^{3} T^{5} + 1147092 p^{6} T^{6} + 628 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 1824 T + 2267302 T^{2} - 1853911234 T^{3} + 1309529072650 T^{4} - 1853911234 p^{3} T^{5} + 2267302 p^{6} T^{6} - 1824 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 1024 T + 1357768 T^{2} + 1056549188 T^{3} + 881935228670 T^{4} + 1056549188 p^{3} T^{5} + 1357768 p^{6} T^{6} + 1024 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 38 T + 828200 T^{2} + 993274186 T^{3} + 126561648798 T^{4} + 993274186 p^{3} T^{5} + 828200 p^{6} T^{6} + 38 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 1512 T + 1002208 T^{2} + 1342237786 T^{3} + 1765886897618 T^{4} + 1342237786 p^{3} T^{5} + 1002208 p^{6} T^{6} + 1512 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 1058 T + 3169588 T^{2} + 2042384262 T^{3} + 3880043990950 T^{4} + 2042384262 p^{3} T^{5} + 3169588 p^{6} T^{6} + 1058 p^{9} T^{7} + p^{12} T^{8} \)
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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

−7.49884509641978569354038698220, −6.97888370324124970288790189933, −6.85620572372765171325539305699, −6.79774113502391321541486222744, −6.72077351920284394885003558375, −6.03778095410398895114241711374, −5.89127398619223946932336641164, −5.86817480846295338259892472753, −5.71330151505878563150964230125, −5.11812217194043363745384366305, −4.97234668681955743617978215349, −4.89007022702133428032997798662, −4.83049813462788273537960047997, −3.93872568100893297277751579581, −3.90808172907497004126669748414, −3.88505805368799704215500218964, −3.45803006894579457936742409880, −2.87402661308833289691089857801, −2.49716072501869510502240162212, −2.44003681715624846230443591989, −2.36776877879929117152725399652, −1.28209659550735849250093696686, −1.22296189582313503932657566873, −1.17698783491237753221250575281, −0.988078825294213372526571754238, 0, 0, 0, 0, 0.988078825294213372526571754238, 1.17698783491237753221250575281, 1.22296189582313503932657566873, 1.28209659550735849250093696686, 2.36776877879929117152725399652, 2.44003681715624846230443591989, 2.49716072501869510502240162212, 2.87402661308833289691089857801, 3.45803006894579457936742409880, 3.88505805368799704215500218964, 3.90808172907497004126669748414, 3.93872568100893297277751579581, 4.83049813462788273537960047997, 4.89007022702133428032997798662, 4.97234668681955743617978215349, 5.11812217194043363745384366305, 5.71330151505878563150964230125, 5.86817480846295338259892472753, 5.89127398619223946932336641164, 6.03778095410398895114241711374, 6.72077351920284394885003558375, 6.79774113502391321541486222744, 6.85620572372765171325539305699, 6.97888370324124970288790189933, 7.49884509641978569354038698220

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