Properties

Label 8-252e4-1.1-c4e4-0-3
Degree $8$
Conductor $4032758016$
Sign $1$
Analytic cond. $460446.$
Root an. cond. $5.10384$
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  + 182·7-s + 144·19-s − 851·25-s + 4.31e3·31-s + 4.52e3·37-s + 7.98e3·43-s + 2.00e4·49-s + 5.46e3·61-s − 5.00e3·67-s + 2.10e4·73-s + 6.60e3·79-s − 2.95e4·103-s + 1.17e4·109-s − 643·121-s + 127-s + 131-s + 2.62e4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.07e5·169-s + 173-s − 1.54e5·175-s + ⋯
L(s)  = 1  + 26/7·7-s + 0.398·19-s − 1.36·25-s + 4.48·31-s + 3.30·37-s + 4.31·43-s + 8.34·49-s + 1.46·61-s − 1.11·67-s + 3.95·73-s + 1.05·79-s − 2.78·103-s + 0.987·109-s − 0.0439·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 1.48·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 3.75·169-s + 3.34e−5·173-s − 5.05·175-s + ⋯

Functional equation

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

Invariants

Degree: \(8\)
Conductor: \(2^{8} \cdot 3^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(460446.\)
Root analytic conductor: \(5.10384\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{8} \cdot 3^{8} \cdot 7^{4} ,\ ( \ : 2, 2, 2, 2 ),\ 1 )\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(18.07919002\)
\(L(\frac12)\) \(\approx\) \(18.07919002\)
\(L(3)\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( ( 1 - 13 p T + p^{4} T^{2} )^{2} \)
good5$C_2^3$ \( 1 + 851 T^{2} + 333576 T^{4} + 851 p^{8} T^{6} + p^{16} T^{8} \)
11$C_2^3$ \( 1 + 643 T^{2} - 213945432 T^{4} + 643 p^{8} T^{6} + p^{16} T^{8} \)
13$C_2^2$ \( ( 1 - 53654 T^{2} + p^{8} T^{4} )^{2} \)
17$C_2^3$ \( 1 + 7442 T^{2} - 6920374077 T^{4} + 7442 p^{8} T^{6} + p^{16} T^{8} \)
19$C_2^2$ \( ( 1 - 72 T + 132049 T^{2} - 72 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
23$C_2^3$ \( 1 + 517618 T^{2} + 189617408643 T^{4} + 517618 p^{8} T^{6} + p^{16} T^{8} \)
29$C_2^2$ \( ( 1 + 1384637 T^{2} + p^{8} T^{4} )^{2} \)
31$C_2^2$ \( ( 1 - 2157 T + 2474404 T^{2} - 2157 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
37$C_2^2$ \( ( 1 - 2264 T + 3251535 T^{2} - 2264 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
41$C_2^2$ \( ( 1 - 4654022 T^{2} + p^{8} T^{4} )^{2} \)
43$C_2$ \( ( 1 - 1996 T + p^{4} T^{2} )^{4} \)
47$C_2^3$ \( 1 - 1771738 T^{2} - 20672231121117 T^{4} - 1771738 p^{8} T^{6} + p^{16} T^{8} \)
53$C_2^3$ \( 1 + 49363 T^{2} - 62257253705592 T^{4} + 49363 p^{8} T^{6} + p^{16} T^{8} \)
59$C_2^3$ \( 1 + 7466747 T^{2} - 91078126842312 T^{4} + 7466747 p^{8} T^{6} + p^{16} T^{8} \)
61$C_2^2$ \( ( 1 - 2730 T + 16330141 T^{2} - 2730 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
67$C_2^2$ \( ( 1 + 2504 T - 13881105 T^{2} + 2504 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
71$C_2^2$ \( ( 1 + 41127662 T^{2} + p^{8} T^{4} )^{2} \)
73$C_2^2$ \( ( 1 - 10548 T + 65485009 T^{2} - 10548 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
79$C_2^2$ \( ( 1 - 3301 T - 28053480 T^{2} - 3301 p^{4} T^{3} + p^{8} T^{4} )^{2} \)
83$C_2^2$ \( ( 1 - 15904667 T^{2} + p^{8} T^{4} )^{2} \)
89$C_2^3$ \( 1 + 21704582 T^{2} - 3465499925907357 T^{4} + 21704582 p^{8} T^{6} + p^{16} T^{8} \)
97$C_2^2$ \( ( 1 - 158532887 T^{2} + p^{8} T^{4} )^{2} \)
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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

−8.103772315253889333946331031077, −7.77358470585450827675407046448, −7.59835159399450244623198505116, −7.54707968757125088097074606952, −7.42804639802878789229982957735, −6.57930774558332380919801667412, −6.36644453978679674158170601891, −6.32609850651249378728135147988, −5.93915464759039468201770129951, −5.33388848665714359172529503485, −5.29836248419932578602422769937, −5.24949868334921910438836512556, −4.57309002218306960097144566991, −4.42039890889123112898026410997, −4.27871244466151222750322580958, −4.09873142176576303743490003312, −3.70191428663502656291562751797, −2.85007160158998462612154065424, −2.46956062512774648759404441341, −2.37175011286258459039706613548, −2.24589057534894982880237087162, −1.39922221014893909661947059908, −1.13052237743742901246179875465, −0.822302193929903503952038767171, −0.74432983752789688250206587120, 0.74432983752789688250206587120, 0.822302193929903503952038767171, 1.13052237743742901246179875465, 1.39922221014893909661947059908, 2.24589057534894982880237087162, 2.37175011286258459039706613548, 2.46956062512774648759404441341, 2.85007160158998462612154065424, 3.70191428663502656291562751797, 4.09873142176576303743490003312, 4.27871244466151222750322580958, 4.42039890889123112898026410997, 4.57309002218306960097144566991, 5.24949868334921910438836512556, 5.29836248419932578602422769937, 5.33388848665714359172529503485, 5.93915464759039468201770129951, 6.32609850651249378728135147988, 6.36644453978679674158170601891, 6.57930774558332380919801667412, 7.42804639802878789229982957735, 7.54707968757125088097074606952, 7.59835159399450244623198505116, 7.77358470585450827675407046448, 8.103772315253889333946331031077

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