Properties

Label 8-825e4-1.1-c3e4-0-0
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $5.61409\times 10^{6}$
Root an. cond. $6.97686$
Motivic weight $3$
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  − 17·4-s − 18·9-s − 44·11-s + 113·16-s − 100·19-s + 396·29-s + 720·31-s + 306·36-s − 1.56e3·41-s + 748·44-s + 308·49-s + 344·59-s − 1.55e3·61-s − 17·64-s + 1.26e3·71-s + 1.70e3·76-s − 1.30e3·79-s + 243·81-s + 1.51e3·89-s + 792·99-s + 1.30e3·101-s − 684·109-s − 6.73e3·116-s + 1.21e3·121-s − 1.22e4·124-s + 127-s + 131-s + ⋯
L(s)  = 1  − 2.12·4-s − 2/3·9-s − 1.20·11-s + 1.76·16-s − 1.20·19-s + 2.53·29-s + 4.17·31-s + 1.41·36-s − 5.95·41-s + 2.56·44-s + 0.897·49-s + 0.759·59-s − 3.26·61-s − 0.0332·64-s + 2.10·71-s + 2.56·76-s − 1.85·79-s + 1/3·81-s + 1.80·89-s + 0.804·99-s + 1.28·101-s − 0.601·109-s − 5.38·116-s + 0.909·121-s − 8.86·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5.61409\times 10^{6}\)
Root analytic conductor: \(6.97686\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.1142235561\)
\(L(\frac12)\) \(\approx\) \(0.1142235561\)
\(L(\frac{5}{2})\) 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)$
bad3$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
5 \( 1 \)
11$C_1$ \( ( 1 + p T )^{4} \)
good2$D_4\times C_2$ \( 1 + 17 T^{2} + 11 p^{4} T^{4} + 17 p^{6} T^{6} + p^{12} T^{8} \)
7$D_4\times C_2$ \( 1 - 44 p T^{2} + 35526 T^{4} - 44 p^{7} T^{6} + p^{12} T^{8} \)
13$D_4\times C_2$ \( 1 - 8144 T^{2} + 26147502 T^{4} - 8144 p^{6} T^{6} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 - 4528 T^{2} - 3874 T^{4} - 4528 p^{6} T^{6} + p^{12} T^{8} \)
19$D_{4}$ \( ( 1 + 50 T + 14246 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 34840 T^{2} + 555984878 T^{4} - 34840 p^{6} T^{6} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 - 198 T + 57706 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 - 360 T + 90430 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 17676 T^{2} - 1844934986 T^{4} - 17676 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 + 782 T + 285970 T^{2} + 782 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 210744 T^{2} + 21303493054 T^{4} - 210744 p^{6} T^{6} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 - 114328 T^{2} + 15430252046 T^{4} - 114328 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 317840 T^{2} + 50316127566 T^{4} - 317840 p^{6} T^{6} + p^{12} T^{8} \)
59$D_{4}$ \( ( 1 - 172 T + 175654 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 778 T + 577250 T^{2} + 778 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 454988 T^{2} + 98091174486 T^{4} - 454988 p^{6} T^{6} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 630 T + 744334 T^{2} - 630 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 678236 T^{2} + 385736611110 T^{4} - 678236 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 + 652 T + 589506 T^{2} + 652 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 1054364 T^{2} + 869850231414 T^{4} - 1054364 p^{6} T^{6} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 - 756 T + 1427110 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 1760636 T^{2} + 2258265821574 T^{4} - 1760636 p^{6} T^{6} + p^{12} T^{8} \)
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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

−6.87090082648338106374639347948, −6.69026628224795576758126242600, −6.44196288977741245650399220509, −6.40352955534580610646980517930, −5.94489986311442455734048861629, −5.61211940066337639298633050077, −5.55016951689399035500583943240, −5.09460488016520987726379059902, −4.96587645635245544888778507983, −4.68908931992931874560240375029, −4.54822864837259315271261709495, −4.46336118178523952277968438202, −4.38865190526260566825593154124, −3.62446093411829798716442687730, −3.56347057291526263799893827187, −3.19057215278539379595302908791, −3.03332406419574166703074924222, −2.59600989687639132322466372467, −2.56682912881532821197533110756, −1.85713428132662860627649155592, −1.81393487503975013913538260158, −1.15557914797694368133464218840, −0.65509718880976434817584810235, −0.63986573779988583038275854889, −0.07088090112064358365883675183, 0.07088090112064358365883675183, 0.63986573779988583038275854889, 0.65509718880976434817584810235, 1.15557914797694368133464218840, 1.81393487503975013913538260158, 1.85713428132662860627649155592, 2.56682912881532821197533110756, 2.59600989687639132322466372467, 3.03332406419574166703074924222, 3.19057215278539379595302908791, 3.56347057291526263799893827187, 3.62446093411829798716442687730, 4.38865190526260566825593154124, 4.46336118178523952277968438202, 4.54822864837259315271261709495, 4.68908931992931874560240375029, 4.96587645635245544888778507983, 5.09460488016520987726379059902, 5.55016951689399035500583943240, 5.61211940066337639298633050077, 5.94489986311442455734048861629, 6.40352955534580610646980517930, 6.44196288977741245650399220509, 6.69026628224795576758126242600, 6.87090082648338106374639347948

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