Properties

Label 4-2160e2-1.1-c3e2-0-1
Degree $4$
Conductor $4665600$
Sign $1$
Analytic cond. $16242.0$
Root an. cond. $11.2891$
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  − 10·5-s + 2·7-s − 42·11-s − 14·13-s + 24·17-s + 140·19-s − 126·23-s + 75·25-s − 126·29-s + 56·31-s − 20·35-s − 572·37-s − 66·41-s + 530·43-s + 396·47-s − 494·49-s − 504·53-s + 420·55-s − 438·59-s − 602·61-s + 140·65-s + 920·67-s − 798·71-s − 770·73-s − 84·77-s + 1.72e3·79-s + 486·83-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.107·7-s − 1.15·11-s − 0.298·13-s + 0.342·17-s + 1.69·19-s − 1.14·23-s + 3/5·25-s − 0.806·29-s + 0.324·31-s − 0.0965·35-s − 2.54·37-s − 0.251·41-s + 1.87·43-s + 1.22·47-s − 1.44·49-s − 1.30·53-s + 1.02·55-s − 0.966·59-s − 1.26·61-s + 0.267·65-s + 1.67·67-s − 1.33·71-s − 1.23·73-s − 0.124·77-s + 2.45·79-s + 0.642·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4665600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4665600\)    =    \(2^{8} \cdot 3^{6} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(16242.0\)
Root analytic conductor: \(11.2891\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4665600,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8166027828\)
\(L(\frac12)\) \(\approx\) \(0.8166027828\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + p T )^{2} \)
good7$D_{4}$ \( 1 - 2 T + 498 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 42 T + 2914 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 14 T - 282 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 24 T + 709 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 140 T + 18429 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 126 T + 16207 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 126 T + 29878 T^{2} + 126 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 56 T + 5745 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 286 T + p^{3} T^{2} )^{2} \)
41$D_{4}$ \( 1 + 66 T + 55582 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 530 T + 219978 T^{2} - 530 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 396 T + 209806 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 504 T + 243133 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 438 T + 449458 T^{2} + 438 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 602 T + 211167 T^{2} + 602 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 920 T + 479730 T^{2} - 920 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 798 T + 832498 T^{2} + 798 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 770 T + 788478 T^{2} + 770 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 1724 T + 1645773 T^{2} - 1724 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 486 T + 1054447 T^{2} - 486 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 294 T + 681406 T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 112 T + 618882 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.749016548482755205982913833402, −8.628442886736332769039368087324, −7.83580942002962935874840310454, −7.77489384298858247854903252706, −7.48002417380488389910445260408, −7.29576781877854795537829382895, −6.48166615628107344836457939849, −6.27366145499345130168390591261, −5.53745967154544629117673218131, −5.43300764934482087129222436746, −4.77455970135566136532238914649, −4.73258208864253161497545598446, −3.77126314787628267190291023131, −3.73650548552533821513075148450, −3.03143125922218684439054559202, −2.82452064750427295288237937925, −2.00073435894270735420874362159, −1.60705514738449979110797926785, −0.815837609785272161945013762832, −0.22924545045305409245544262260, 0.22924545045305409245544262260, 0.815837609785272161945013762832, 1.60705514738449979110797926785, 2.00073435894270735420874362159, 2.82452064750427295288237937925, 3.03143125922218684439054559202, 3.73650548552533821513075148450, 3.77126314787628267190291023131, 4.73258208864253161497545598446, 4.77455970135566136532238914649, 5.43300764934482087129222436746, 5.53745967154544629117673218131, 6.27366145499345130168390591261, 6.48166615628107344836457939849, 7.29576781877854795537829382895, 7.48002417380488389910445260408, 7.77489384298858247854903252706, 7.83580942002962935874840310454, 8.628442886736332769039368087324, 8.749016548482755205982913833402

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