Properties

Label 8-960e4-1.1-c3e4-0-2
Degree $8$
Conductor $849346560000$
Sign $1$
Analytic cond. $1.02931\times 10^{7}$
Root an. cond. $7.52607$
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  − 6·5-s − 18·9-s − 84·11-s − 112·19-s + 146·25-s − 636·29-s − 104·31-s − 816·41-s + 108·45-s + 616·49-s + 504·55-s + 372·59-s − 680·61-s + 72·71-s + 760·79-s + 243·81-s − 2.23e3·89-s + 672·95-s + 1.51e3·99-s + 2.24e3·101-s − 1.32e3·109-s − 176·121-s − 2.28e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 0.536·5-s − 2/3·9-s − 2.30·11-s − 1.35·19-s + 1.16·25-s − 4.07·29-s − 0.602·31-s − 3.10·41-s + 0.357·45-s + 1.79·49-s + 1.23·55-s + 0.820·59-s − 1.42·61-s + 0.120·71-s + 1.08·79-s + 1/3·81-s − 2.65·89-s + 0.725·95-s + 1.53·99-s + 2.21·101-s − 1.16·109-s − 0.132·121-s − 1.63·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{24} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1.02931\times 10^{7}\)
Root analytic conductor: \(7.52607\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{24} \cdot 3^{4} \cdot 5^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(0.1124356917\)
\(L(\frac12)\) \(\approx\) \(0.1124356917\)
\(L(\frac{5}{2})\) 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 \( 1 \)
3$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
5$C_2^2$ \( 1 + 6 T - 22 p T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
good7$D_4\times C_2$ \( 1 - 88 p T^{2} + 316878 T^{4} - 88 p^{7} T^{6} + p^{12} T^{8} \)
11$D_{4}$ \( ( 1 + 42 T + 2734 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
13$D_4\times C_2$ \( 1 - 5008 T^{2} + 13678638 T^{4} - 5008 p^{6} T^{6} + p^{12} T^{8} \)
17$D_4\times C_2$ \( 1 - 12400 T^{2} + 76051038 T^{4} - 12400 p^{6} T^{6} + p^{12} T^{8} \)
19$D_{4}$ \( ( 1 + 56 T + 8598 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
23$D_4\times C_2$ \( 1 - 46204 T^{2} + 828262758 T^{4} - 46204 p^{6} T^{6} + p^{12} T^{8} \)
29$D_{4}$ \( ( 1 + 318 T + 55978 T^{2} + 318 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
31$D_{4}$ \( ( 1 + 52 T + 58782 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
37$D_4\times C_2$ \( 1 - 96016 T^{2} + 4637534478 T^{4} - 96016 p^{6} T^{6} + p^{12} T^{8} \)
41$D_{4}$ \( ( 1 + 408 T + 177982 T^{2} + 408 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
43$D_4\times C_2$ \( 1 - 121900 T^{2} + 14997346998 T^{4} - 121900 p^{6} T^{6} + p^{12} T^{8} \)
47$D_4\times C_2$ \( 1 - 225580 T^{2} + 31469120358 T^{4} - 225580 p^{6} T^{6} + p^{12} T^{8} \)
53$D_4\times C_2$ \( 1 - 376864 T^{2} + 68697431598 T^{4} - 376864 p^{6} T^{6} + p^{12} T^{8} \)
59$D_{4}$ \( ( 1 - 186 T + 419038 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
61$D_{4}$ \( ( 1 + 340 T + 388398 T^{2} + 340 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
67$D_4\times C_2$ \( 1 - 861340 T^{2} + 346621507638 T^{4} - 861340 p^{6} T^{6} + p^{12} T^{8} \)
71$D_{4}$ \( ( 1 - 36 T + 384046 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
73$D_4\times C_2$ \( 1 - 429844 T^{2} + 136740794118 T^{4} - 429844 p^{6} T^{6} + p^{12} T^{8} \)
79$D_{4}$ \( ( 1 - 380 T + 99678 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
83$D_4\times C_2$ \( 1 - 916108 T^{2} + 434315280918 T^{4} - 916108 p^{6} T^{6} + p^{12} T^{8} \)
89$D_{4}$ \( ( 1 + 1116 T + 1508758 T^{2} + 1116 p^{3} T^{3} + p^{6} T^{4} )^{2} \)
97$D_4\times C_2$ \( 1 - 2174980 T^{2} + 2500346420358 T^{4} - 2174980 p^{6} T^{6} + p^{12} T^{8} \)
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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

−6.62612940830560114252244471088, −6.60461857970188157464548023136, −6.60169732000237952391599215220, −5.91794218707413854285266948417, −5.77135411216349948249839241662, −5.63703641658160895088353320896, −5.30778269588247831751305868736, −5.19805131087996443757181601055, −5.16172532638493311697288665766, −4.63897666143510660975661492301, −4.59883035664238473308175217030, −3.97110719126549774277080327935, −3.85553622034528379319277810257, −3.77553096644257448858535320743, −3.52424259675361632723343513281, −2.89842614835947036478143944502, −2.80682396021086534671052688076, −2.80073767316836507443678076121, −2.17651159235255094976936830490, −1.95616523775568600498664745670, −1.80484286687918247020763659406, −1.39653221423124725637504785611, −0.810534400152199493417260728516, −0.25631775832331570205085822254, −0.10466918239197473057441049062, 0.10466918239197473057441049062, 0.25631775832331570205085822254, 0.810534400152199493417260728516, 1.39653221423124725637504785611, 1.80484286687918247020763659406, 1.95616523775568600498664745670, 2.17651159235255094976936830490, 2.80073767316836507443678076121, 2.80682396021086534671052688076, 2.89842614835947036478143944502, 3.52424259675361632723343513281, 3.77553096644257448858535320743, 3.85553622034528379319277810257, 3.97110719126549774277080327935, 4.59883035664238473308175217030, 4.63897666143510660975661492301, 5.16172532638493311697288665766, 5.19805131087996443757181601055, 5.30778269588247831751305868736, 5.63703641658160895088353320896, 5.77135411216349948249839241662, 5.91794218707413854285266948417, 6.60169732000237952391599215220, 6.60461857970188157464548023136, 6.62612940830560114252244471088

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