Properties

Label 4-1008e2-1.1-c5e2-0-9
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $26136.1$
Root an. cond. $12.7148$
Motivic weight $5$
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  − 48·5-s + 98·7-s + 368·11-s − 156·13-s + 3.31e3·17-s − 3.73e3·19-s + 1.55e3·23-s − 822·25-s − 1.72e3·29-s + 3.62e3·31-s − 4.70e3·35-s + 6.99e3·37-s − 2.61e4·41-s + 3.01e4·43-s − 1.14e4·47-s + 7.20e3·49-s + 1.73e4·53-s − 1.76e4·55-s + 1.50e4·59-s + 3.55e4·61-s + 7.48e3·65-s + 7.05e4·67-s + 4.07e4·71-s − 5.38e4·73-s + 3.60e4·77-s + 9.74e3·79-s + 3.13e4·83-s + ⋯
L(s)  = 1  − 0.858·5-s + 0.755·7-s + 0.916·11-s − 0.256·13-s + 2.77·17-s − 2.37·19-s + 0.611·23-s − 0.263·25-s − 0.381·29-s + 0.677·31-s − 0.649·35-s + 0.840·37-s − 2.43·41-s + 2.48·43-s − 0.754·47-s + 3/7·49-s + 0.849·53-s − 0.787·55-s + 0.561·59-s + 1.22·61-s + 0.219·65-s + 1.91·67-s + 0.959·71-s − 1.18·73-s + 0.693·77-s + 0.175·79-s + 0.499·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26136.1\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.597419059\)
\(L(\frac12)\) \(\approx\) \(4.597419059\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 - p^{2} T )^{2} \)
good5$D_{4}$ \( 1 + 48 T + 3126 T^{2} + 48 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 368 T + 153346 T^{2} - 368 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 12 p T - 198530 T^{2} + 12 p^{6} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 3312 T + 5420878 T^{2} - 3312 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 3736 T + 8100630 T^{2} + 3736 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 1552 T + 13432090 T^{2} - 1552 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 1728 T + 36312922 T^{2} + 1728 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 3624 T + 60503758 T^{2} - 3624 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 6996 T + 149067406 T^{2} - 6996 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 26160 T + 385589214 T^{2} + 26160 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 30184 T + 521633798 T^{2} - 30184 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 11424 T + 490819086 T^{2} + 11424 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 17376 T + 113903098 T^{2} - 17376 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 15008 T + 1001295814 T^{2} - 15008 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 35564 T + 1803486974 T^{2} - 35564 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 70504 T + 3904156406 T^{2} - 70504 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 40752 T + 3146114778 T^{2} - 40752 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 53892 T + 4664755414 T^{2} + 53892 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 9744 T + 5082885982 T^{2} - 9744 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 31360 T - 880376314 T^{2} - 31360 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 35952 T + 5931688606 T^{2} + 35952 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 66652 T - 1905212410 T^{2} - 66652 p^{5} T^{3} + p^{10} T^{4} \)
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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

−9.437178490812495738501778184547, −8.937696915065360611865023181456, −8.388192975279718331297712245177, −8.307752911442693177623651113460, −7.71901548135477744489901295616, −7.59071692144348407450191057573, −6.78168649375959368488112902176, −6.70982156890324907684109681449, −5.97690695588593766924091235411, −5.56214737534156306547799842406, −5.13630493293211809642447845212, −4.58424874277074259747280112828, −3.98577942009051701596295020008, −3.88604499012928880278022627358, −3.28175071461452794460888799997, −2.63889989792800366198610611361, −1.99328476503085769737991578355, −1.51766111134679628836487085585, −0.71170881430580796224315443729, −0.60912811474838416043248964761, 0.60912811474838416043248964761, 0.71170881430580796224315443729, 1.51766111134679628836487085585, 1.99328476503085769737991578355, 2.63889989792800366198610611361, 3.28175071461452794460888799997, 3.88604499012928880278022627358, 3.98577942009051701596295020008, 4.58424874277074259747280112828, 5.13630493293211809642447845212, 5.56214737534156306547799842406, 5.97690695588593766924091235411, 6.70982156890324907684109681449, 6.78168649375959368488112902176, 7.59071692144348407450191057573, 7.71901548135477744489901295616, 8.307752911442693177623651113460, 8.388192975279718331297712245177, 8.937696915065360611865023181456, 9.437178490812495738501778184547

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