Properties

Label 4-28e4-1.1-c3e2-0-5
Degree $4$
Conductor $614656$
Sign $1$
Analytic cond. $2139.75$
Root an. cond. $6.80128$
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  − 14·3-s + 93·9-s + 238·19-s + 7·25-s − 238·27-s + 420·29-s + 602·31-s − 154·37-s − 714·47-s + 654·53-s − 3.33e3·57-s + 1.21e3·59-s − 98·75-s − 1.68e3·81-s − 1.17e3·83-s − 5.88e3·87-s − 8.42e3·93-s − 2.00e3·103-s + 1.75e3·109-s + 2.15e3·111-s + 108·113-s + 2.51e3·121-s + 127-s + 131-s + 137-s + 139-s + 9.99e3·141-s + ⋯
L(s)  = 1  − 2.69·3-s + 31/9·9-s + 2.87·19-s + 0.0559·25-s − 1.69·27-s + 2.68·29-s + 3.48·31-s − 0.684·37-s − 2.21·47-s + 1.69·53-s − 7.74·57-s + 2.68·59-s − 0.150·75-s − 2.31·81-s − 1.55·83-s − 7.24·87-s − 9.39·93-s − 1.91·103-s + 1.53·109-s + 1.84·111-s + 0.0899·113-s + 1.88·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 5.97·141-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2139.75\)
Root analytic conductor: \(6.80128\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 614656,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.278606774\)
\(L(\frac12)\) \(\approx\) \(1.278606774\)
\(L(\frac{5}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2$ \( ( 1 + 7 T + p^{3} T^{2} )^{2} \)
5$C_2^2$ \( 1 - 7 T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 2515 T^{2} + p^{6} T^{4} \)
13$C_2^2$ \( 1 + 406 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 8743 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 119 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6547 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 210 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 301 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 77 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 123970 T^{2} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 144314 T^{2} + p^{6} T^{4} \)
47$C_2$ \( ( 1 + 357 T + p^{3} T^{2} )^{2} \)
53$C_2$ \( ( 1 - 327 T + p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 - 609 T + p^{3} T^{2} )^{2} \)
61$C_2^2$ \( 1 + 18865 T^{2} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 576683 T^{2} + p^{6} T^{4} \)
71$C_2^2$ \( 1 - 668194 T^{2} + p^{6} T^{4} \)
73$C_2^2$ \( 1 - 774767 T^{2} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 326195 T^{2} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 588 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 868063 T^{2} + p^{6} T^{4} \)
97$C_2^2$ \( 1 - 1728146 T^{2} + p^{6} T^{4} \)
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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

−10.31260916158151608454087958680, −9.856793805662405199965788384176, −9.781245614838162563162547064040, −8.613951454927246511891886340641, −8.493826541773049244036182601605, −7.975759585656567294373869990103, −7.21051485179773525147386684721, −6.76098075551385359811793984362, −6.70185711502962838766402788443, −5.97752351647935139736548494839, −5.82897515418733790385485262768, −5.04531806695924021001080301759, −4.99912271917732708749787271509, −4.66319488744134416585174614863, −3.85147138565348271921374541197, −2.94775865919182988725409765903, −2.71018666374217934530103932717, −1.16263738638319280775398525990, −1.03744820017491404313415626951, −0.51776287910041232079874345725, 0.51776287910041232079874345725, 1.03744820017491404313415626951, 1.16263738638319280775398525990, 2.71018666374217934530103932717, 2.94775865919182988725409765903, 3.85147138565348271921374541197, 4.66319488744134416585174614863, 4.99912271917732708749787271509, 5.04531806695924021001080301759, 5.82897515418733790385485262768, 5.97752351647935139736548494839, 6.70185711502962838766402788443, 6.76098075551385359811793984362, 7.21051485179773525147386684721, 7.975759585656567294373869990103, 8.493826541773049244036182601605, 8.613951454927246511891886340641, 9.781245614838162563162547064040, 9.856793805662405199965788384176, 10.31260916158151608454087958680

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