Properties

Label 8-882e4-1.1-c4e4-0-3
Degree $8$
Conductor $605165749776$
Sign $1$
Analytic cond. $6.90958\times 10^{7}$
Root an. cond. $9.54841$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·4-s − 88·13-s + 192·16-s + 1.00e3·19-s + 2.37e3·25-s + 1.01e3·31-s − 928·37-s − 592·43-s + 1.40e3·52-s + 1.25e4·61-s − 2.04e3·64-s + 1.68e4·67-s + 6.00e3·73-s − 1.60e4·76-s − 1.58e4·79-s − 5.27e4·97-s − 3.79e4·100-s + 3.09e4·103-s + 2.00e4·109-s − 7.40e3·121-s − 1.62e4·124-s + 127-s + 131-s + 137-s + 139-s + 1.48e4·148-s + 149-s + ⋯
L(s)  = 1  − 4-s − 0.520·13-s + 3/4·16-s + 2.77·19-s + 3.79·25-s + 1.05·31-s − 0.677·37-s − 0.320·43-s + 0.520·52-s + 3.37·61-s − 1/2·64-s + 3.75·67-s + 1.12·73-s − 2.77·76-s − 2.53·79-s − 5.60·97-s − 3.79·100-s + 2.91·103-s + 1.68·109-s − 0.505·121-s − 1.05·124-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 0.677·148-s + 4.50e−5·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(6.90958\times 10^{7}\)
Root analytic conductor: \(9.54841\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(5.140868267\)
\(L(\frac12)\) \(\approx\) \(5.140868267\)
\(L(3)\) 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$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$D_4\times C_2$ \( 1 - 2372 T^{2} + 2185046 T^{4} - 2372 p^{8} T^{6} + p^{16} T^{8} \)
11$D_4\times C_2$ \( 1 + 7408 T^{2} + 44053826 T^{4} + 7408 p^{8} T^{6} + p^{16} T^{8} \)
13$D_{4}$ \( ( 1 + 44 T + 54806 T^{2} + 44 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
17$D_4\times C_2$ \( 1 - 155492 T^{2} + 19324224086 T^{4} - 155492 p^{8} T^{6} + p^{16} T^{8} \)
19$D_{4}$ \( ( 1 - 500 T + 212010 T^{2} - 500 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 507248 T^{2} + 194115433730 T^{4} - 507248 p^{8} T^{6} + p^{16} T^{8} \)
29$D_4\times C_2$ \( 1 - 2696528 T^{2} + 2815621714370 T^{4} - 2696528 p^{8} T^{6} + p^{16} T^{8} \)
31$D_{4}$ \( ( 1 - 508 T + 1273130 T^{2} - 508 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
37$D_{4}$ \( ( 1 + 464 T + 3357618 T^{2} + 464 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
41$D_4\times C_2$ \( 1 - 10422916 T^{2} + 43126586993814 T^{4} - 10422916 p^{8} T^{6} + p^{16} T^{8} \)
43$D_{4}$ \( ( 1 + 296 T + 6229506 T^{2} + 296 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
47$D_4\times C_2$ \( 1 - 3843028 T^{2} + 610858828518 T^{4} - 3843028 p^{8} T^{6} + p^{16} T^{8} \)
53$D_4\times C_2$ \( 1 - 31511200 T^{2} + 372757754218434 T^{4} - 31511200 p^{8} T^{6} + p^{16} T^{8} \)
59$D_4\times C_2$ \( 1 - 1937300 T^{2} + 227869699778342 T^{4} - 1937300 p^{8} T^{6} + p^{16} T^{8} \)
61$D_{4}$ \( ( 1 - 6280 T + 26046614 T^{2} - 6280 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
67$D_{4}$ \( ( 1 - 8436 T + 38430934 T^{2} - 8436 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
71$D_4\times C_2$ \( 1 - 60749936 T^{2} + 2213591693878274 T^{4} - 60749936 p^{8} T^{6} + p^{16} T^{8} \)
73$D_{4}$ \( ( 1 - 3000 T + 33509782 T^{2} - 3000 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
79$D_{4}$ \( ( 1 + 100 p T + 36724934 T^{2} + 100 p^{5} T^{3} + p^{8} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 181804868 T^{2} + 12763622956082630 T^{4} - 181804868 p^{8} T^{6} + p^{16} T^{8} \)
89$D_4\times C_2$ \( 1 - 28212772 T^{2} - 86397249754794 T^{4} - 28212772 p^{8} T^{6} + p^{16} T^{8} \)
97$D_{4}$ \( ( 1 + 26352 T + 344559046 T^{2} + 26352 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
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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

−6.82501385968950738782697140404, −6.44064727774483697361553891513, −6.43442253068179621948198835280, −5.73571829076767211515930098431, −5.64830527078146692665911329754, −5.42903809438056250638053347414, −5.37714588936409782614776114913, −4.85112428395428128515140969742, −4.77620405399186524621303411599, −4.68958176869050065678287998193, −4.68691317082859146152073645470, −3.72075107952421561131680259639, −3.68722673859346565565622407423, −3.66827877636530377762938067680, −3.54111732035146710650981736431, −2.83467467498851186163768288157, −2.59451707367834560059413640065, −2.54244231000698831662816658739, −2.50205045486187612434946063663, −1.58849870156152123070853479151, −1.28689769883483842379758319764, −1.10152032957770350248117278322, −0.981153627433702129709948396962, −0.54715381176496758590053891282, −0.29185885125697726072869887091, 0.29185885125697726072869887091, 0.54715381176496758590053891282, 0.981153627433702129709948396962, 1.10152032957770350248117278322, 1.28689769883483842379758319764, 1.58849870156152123070853479151, 2.50205045486187612434946063663, 2.54244231000698831662816658739, 2.59451707367834560059413640065, 2.83467467498851186163768288157, 3.54111732035146710650981736431, 3.66827877636530377762938067680, 3.68722673859346565565622407423, 3.72075107952421561131680259639, 4.68691317082859146152073645470, 4.68958176869050065678287998193, 4.77620405399186524621303411599, 4.85112428395428128515140969742, 5.37714588936409782614776114913, 5.42903809438056250638053347414, 5.64830527078146692665911329754, 5.73571829076767211515930098431, 6.43442253068179621948198835280, 6.44064727774483697361553891513, 6.82501385968950738782697140404

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