Properties

Label 4-1008e2-1.1-c5e2-0-12
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 76·5-s − 98·7-s + 564·11-s + 516·13-s + 76·17-s − 2.36e3·19-s − 2.03e3·23-s + 1.89e3·25-s − 8.32e3·29-s + 6.28e3·31-s + 7.44e3·35-s − 2.46e3·37-s + 1.43e4·41-s − 2.31e4·43-s + 1.27e4·47-s + 7.20e3·49-s + 9.89e3·53-s − 4.28e4·55-s + 6.08e4·59-s + 1.61e4·61-s − 3.92e4·65-s + 6.25e4·67-s − 732·71-s + 1.24e3·73-s − 5.52e4·77-s − 1.16e4·79-s + 1.32e5·83-s + ⋯
L(s)  = 1  − 1.35·5-s − 0.755·7-s + 1.40·11-s + 0.846·13-s + 0.0637·17-s − 1.49·19-s − 0.802·23-s + 0.607·25-s − 1.83·29-s + 1.17·31-s + 1.02·35-s − 0.295·37-s + 1.32·41-s − 1.90·43-s + 0.839·47-s + 3/7·49-s + 0.483·53-s − 1.91·55-s + 2.27·59-s + 0.556·61-s − 1.15·65-s + 1.70·67-s − 0.0172·71-s + 0.0273·73-s − 1.06·77-s − 0.209·79-s + 2.10·83-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: \(2\)
Selberg data: \((4,\ 1016064,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$D_{4}$ \( 1 + 76 T + 3878 T^{2} + 76 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 564 T + 306226 T^{2} - 564 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 516 T + 748094 T^{2} - 516 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 76 T + 2532062 T^{2} - 76 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 2360 T + 6283542 T^{2} + 2360 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 2036 T + 7494314 T^{2} + 2036 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 8320 T + 56801498 T^{2} + 8320 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 6280 T + 40192206 T^{2} - 6280 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 2460 T + 126463214 T^{2} + 2460 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 14308 T + 112999982 T^{2} - 14308 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 23128 T + 423835398 T^{2} + 23128 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 12712 T + 311021006 T^{2} - 12712 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 9896 T + 858294074 T^{2} - 9896 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 1032 p T + 2207083270 T^{2} - 1032 p^{6} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 16172 T - 115264002 T^{2} - 16172 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 62568 T + 3672833270 T^{2} - 62568 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 732 T + 3302973034 T^{2} + 732 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 1244 T + 3694959894 T^{2} - 1244 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 11600 T - 2536172706 T^{2} + 11600 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 132288 T + 12228687622 T^{2} - 132288 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 93452 T + 11465536430 T^{2} + 93452 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 272932 T + 35426184966 T^{2} + 272932 p^{5} T^{3} + p^{10} T^{4} \)
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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

−8.785474333218249660029969749793, −8.707695851040274968462818739396, −8.198614110595789713326933444586, −7.889181730772941489060559988848, −7.31991522860471513469995861811, −6.84307016204405697481705717351, −6.42018979964102473048247054500, −6.35868999422999714726884068100, −5.54685942017412267329506734421, −5.23460392932670371498539433887, −4.17802542509806495622393854888, −4.16970663177235272119959208529, −3.63526976356886788167552046243, −3.62105351646575228296849764294, −2.55617567203256503166053996631, −2.19988207503821560395304778043, −1.33707497323353954424246777051, −0.915787900310779323669556340036, 0, 0, 0.915787900310779323669556340036, 1.33707497323353954424246777051, 2.19988207503821560395304778043, 2.55617567203256503166053996631, 3.62105351646575228296849764294, 3.63526976356886788167552046243, 4.16970663177235272119959208529, 4.17802542509806495622393854888, 5.23460392932670371498539433887, 5.54685942017412267329506734421, 6.35868999422999714726884068100, 6.42018979964102473048247054500, 6.84307016204405697481705717351, 7.31991522860471513469995861811, 7.889181730772941489060559988848, 8.198614110595789713326933444586, 8.707695851040274968462818739396, 8.785474333218249660029969749793

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