Properties

Label 4-720e2-1.1-c5e2-0-10
Degree $4$
Conductor $518400$
Sign $1$
Analytic cond. $13334.7$
Root an. cond. $10.7459$
Motivic weight $5$
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  + 90·5-s + 504·11-s + 440·19-s + 4.97e3·25-s + 1.38e4·29-s − 1.35e4·31-s + 396·41-s + 3.00e4·49-s + 4.53e4·55-s − 4.93e4·59-s − 1.13e4·61-s + 1.06e5·71-s − 1.03e5·79-s + 1.99e4·89-s + 3.96e4·95-s + 2.18e5·101-s − 4.20e4·109-s − 1.31e5·121-s + 1.66e5·125-s + 127-s + 131-s + 137-s + 139-s + 1.24e6·145-s + 149-s + 151-s − 1.21e6·155-s + ⋯
L(s)  = 1  + 1.60·5-s + 1.25·11-s + 0.279·19-s + 1.59·25-s + 3.06·29-s − 2.52·31-s + 0.0367·41-s + 1.78·49-s + 2.02·55-s − 1.84·59-s − 0.392·61-s + 2.51·71-s − 1.87·79-s + 0.267·89-s + 0.450·95-s + 2.12·101-s − 0.338·109-s − 0.817·121-s + 0.953·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 4.92·145-s + 3.69e−6·149-s + 3.56e−6·151-s − 4.06·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(518400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(13334.7\)
Root analytic conductor: \(10.7459\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 518400,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(7.124533640\)
\(L(\frac12)\) \(\approx\) \(7.124533640\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 - 18 p T + p^{5} T^{2} \)
good7$C_2^2$ \( 1 - 30050 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 252 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 728330 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 2363810 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 220 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6946370 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 6930 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 6752 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 56462470 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 198 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 293842250 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 347593490 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 802472090 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 24660 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 5698 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 795787610 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 53352 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 883886830 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 51920 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 4053674810 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 9990 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 6923133890 T^{2} + p^{10} T^{4} \)
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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

−9.700666523348687745043247813654, −9.509824879243594892779965412343, −8.946115008338592699635025416803, −8.879766078992221920969633669209, −8.256038195068743085480254198922, −7.67347178334648190997557781558, −7.00659471411899864195927974180, −6.84166133765982907924657219422, −6.17321540506611517345470524712, −6.06745389451677500228793340953, −5.36520414027103962772436912856, −5.09067116896755134092841099086, −4.35770946052570680965135261146, −4.00282453510819375127940587782, −3.11554473428117728236681126926, −2.88818867417723993043865758414, −1.89354746575714583441968555692, −1.84908026162384818924121049253, −1.00945538302512885674216601645, −0.62026114143845586746120146031, 0.62026114143845586746120146031, 1.00945538302512885674216601645, 1.84908026162384818924121049253, 1.89354746575714583441968555692, 2.88818867417723993043865758414, 3.11554473428117728236681126926, 4.00282453510819375127940587782, 4.35770946052570680965135261146, 5.09067116896755134092841099086, 5.36520414027103962772436912856, 6.06745389451677500228793340953, 6.17321540506611517345470524712, 6.84166133765982907924657219422, 7.00659471411899864195927974180, 7.67347178334648190997557781558, 8.256038195068743085480254198922, 8.879766078992221920969633669209, 8.946115008338592699635025416803, 9.509824879243594892779965412343, 9.700666523348687745043247813654

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