Properties

Label 4-528e2-1.1-c3e2-0-3
Degree $4$
Conductor $278784$
Sign $1$
Analytic cond. $970.509$
Root an. cond. $5.58148$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s − 6·5-s − 22·7-s + 27·9-s + 22·11-s + 94·13-s + 36·15-s + 56·17-s − 76·19-s + 132·21-s − 54·23-s − 38·25-s − 108·27-s + 104·29-s − 224·31-s − 132·33-s + 132·35-s − 68·37-s − 564·39-s − 300·41-s − 456·43-s − 162·45-s + 22·47-s − 138·49-s − 336·51-s − 110·53-s − 132·55-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.536·5-s − 1.18·7-s + 9-s + 0.603·11-s + 2.00·13-s + 0.619·15-s + 0.798·17-s − 0.917·19-s + 1.37·21-s − 0.489·23-s − 0.303·25-s − 0.769·27-s + 0.665·29-s − 1.29·31-s − 0.696·33-s + 0.637·35-s − 0.302·37-s − 2.31·39-s − 1.14·41-s − 1.61·43-s − 0.536·45-s + 0.0682·47-s − 0.402·49-s − 0.922·51-s − 0.285·53-s − 0.323·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(970.509\)
Root analytic conductor: \(5.58148\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 278784,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 + p T )^{2} \)
11$C_1$ \( ( 1 - p T )^{2} \)
good5$D_{4}$ \( 1 + 6 T + 74 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 22 T + 622 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 94 T + 6418 T^{2} - 94 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 56 T + 3950 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 4 p T + 8502 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 54 T + 20438 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 104 T + 50742 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 224 T + 60286 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 68 T + 55102 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 300 T + 157382 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 456 T + 208038 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 22 T + 72902 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 110 T + 74154 T^{2} + 110 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 68 T + 322374 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 54 T + 454506 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 1560 T + 1198086 T^{2} + 1560 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 274 T + 692966 T^{2} - 274 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 548 T + 663670 T^{2} - 548 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 922 T + 1100734 T^{2} + 922 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 1184 T + 1135878 T^{2} + 1184 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 956 T + 679382 T^{2} + 956 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 672 T + 1031742 T^{2} + 672 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.33747037209744513795389478573, −9.797961990876120762036885554913, −9.516619691362578128560415445946, −8.879176189338729599418245404816, −8.304900050754115405244181081895, −8.201546517128229264294815509173, −7.29871061589324178307670959481, −6.88042271665584977331907013299, −6.36521487875750320411574650426, −6.26594107574538264427952623040, −5.60199341655892652126726211043, −5.25336137104605056303269658574, −4.22699465150246687628066175577, −4.08205045791384503366025636580, −3.41491125635285612225180576427, −3.01946415186091212137541160255, −1.58572855611915704273670817905, −1.34397948421644746619550212051, 0, 0, 1.34397948421644746619550212051, 1.58572855611915704273670817905, 3.01946415186091212137541160255, 3.41491125635285612225180576427, 4.08205045791384503366025636580, 4.22699465150246687628066175577, 5.25336137104605056303269658574, 5.60199341655892652126726211043, 6.26594107574538264427952623040, 6.36521487875750320411574650426, 6.88042271665584977331907013299, 7.29871061589324178307670959481, 8.201546517128229264294815509173, 8.304900050754115405244181081895, 8.879176189338729599418245404816, 9.516619691362578128560415445946, 9.797961990876120762036885554913, 10.33747037209744513795389478573

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