Properties

Label 4-810e2-1.1-c3e2-0-28
Degree $4$
Conductor $656100$
Sign $1$
Analytic cond. $2284.03$
Root an. cond. $6.91314$
Motivic weight $3$
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  + 4·2-s + 12·4-s + 10·5-s − 5·7-s + 32·8-s + 40·10-s + 51·11-s + 19·13-s − 20·14-s + 80·16-s + 66·17-s − 32·19-s + 120·20-s + 204·22-s − 9·23-s + 75·25-s + 76·26-s − 60·28-s + 255·29-s + 121·31-s + 192·32-s + 264·34-s − 50·35-s + 160·37-s − 128·38-s + 320·40-s − 180·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.269·7-s + 1.41·8-s + 1.26·10-s + 1.39·11-s + 0.405·13-s − 0.381·14-s + 5/4·16-s + 0.941·17-s − 0.386·19-s + 1.34·20-s + 1.97·22-s − 0.0815·23-s + 3/5·25-s + 0.573·26-s − 0.404·28-s + 1.63·29-s + 0.701·31-s + 1.06·32-s + 1.33·34-s − 0.241·35-s + 0.710·37-s − 0.546·38-s + 1.26·40-s − 0.685·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(656100\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2284.03\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 656100,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(14.72101660\)
\(L(\frac12)\) \(\approx\) \(14.72101660\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 - p T )^{2} \)
good7$D_{4}$ \( 1 + 5 T - 78 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 51 T + 2542 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 19 T + 3714 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 66 T + 7834 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 32 T + 10893 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 9 T + 23584 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 255 T + 45778 T^{2} - 255 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 121 T + 43986 T^{2} - 121 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 160 T - 3210 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 180 T + 118213 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 122 T + 159654 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 93 T + 116608 T^{2} - 93 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 771 T + 408622 T^{2} - 771 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 456 T + 311773 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 674 T + 539802 T^{2} + 674 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 1450 T + 1099422 T^{2} - 1450 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1821 T + 1537900 T^{2} - 1821 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 766 T + 675162 T^{2} - 766 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 326 T - 346074 T^{2} + 326 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 1476 T + 1638922 T^{2} + 1476 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1749 T + 2167756 T^{2} - 1749 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 26 T - 1135326 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.07109866762723358005642499396, −9.778706420177548075646046810833, −9.331152158064629367817215885910, −8.845990121460208682760595370720, −8.160156734348242515394846113825, −8.093212285450285519406377526284, −7.17528097923041242047815878428, −6.76750386590071062627606213637, −6.44238199192755266235030928270, −6.21316003066724808107467489577, −5.58463193738174915941298665230, −5.27411635674589691998392541947, −4.61081346287266149214098627517, −4.26973448152753496873960330907, −3.46212354838884293146145306593, −3.44409861927035538703786955925, −2.46493373956719434157691074769, −2.17592194290929759388096540505, −1.19587487348495206675064075555, −0.926624092593186740422253110892, 0.926624092593186740422253110892, 1.19587487348495206675064075555, 2.17592194290929759388096540505, 2.46493373956719434157691074769, 3.44409861927035538703786955925, 3.46212354838884293146145306593, 4.26973448152753496873960330907, 4.61081346287266149214098627517, 5.27411635674589691998392541947, 5.58463193738174915941298665230, 6.21316003066724808107467489577, 6.44238199192755266235030928270, 6.76750386590071062627606213637, 7.17528097923041242047815878428, 8.093212285450285519406377526284, 8.160156734348242515394846113825, 8.845990121460208682760595370720, 9.331152158064629367817215885910, 9.778706420177548075646046810833, 10.07109866762723358005642499396

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