Properties

Label 4-48e2-1.1-c5e2-0-1
Degree $4$
Conductor $2304$
Sign $1$
Analytic cond. $59.2657$
Root an. cond. $2.77460$
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  + 24·3-s + 333·9-s + 1.29e3·11-s − 484·13-s + 2.59e3·23-s − 86·25-s + 2.16e3·27-s + 3.11e4·33-s − 2.41e4·37-s − 1.16e4·39-s + 2.59e4·47-s + 1.53e3·49-s + 1.68e4·59-s − 5.15e4·61-s + 6.22e4·69-s + 1.11e5·71-s + 5.20e4·73-s − 2.06e3·75-s − 2.90e4·81-s − 1.56e5·83-s + 2.06e5·97-s + 4.31e5·99-s + 4.27e4·107-s + 1.17e5·109-s − 5.78e5·111-s − 1.61e5·117-s + 9.37e5·121-s + ⋯
L(s)  = 1  + 1.53·3-s + 1.37·9-s + 3.22·11-s − 0.794·13-s + 1.02·23-s − 0.0275·25-s + 0.570·27-s + 4.97·33-s − 2.89·37-s − 1.22·39-s + 1.71·47-s + 0.0915·49-s + 0.630·59-s − 1.77·61-s + 1.57·69-s + 2.62·71-s + 1.14·73-s − 0.0423·75-s − 0.492·81-s − 2.49·83-s + 2.22·97-s + 4.42·99-s + 0.361·107-s + 0.949·109-s − 4.45·111-s − 1.08·117-s + 5.82·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(59.2657\)
Root analytic conductor: \(2.77460\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2304,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.575240896\)
\(L(\frac12)\) \(\approx\) \(4.575240896\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 - 8 p T + p^{5} T^{2} \)
good5$C_2^2$ \( 1 + 86 T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 - 1538 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 648 T + p^{5} T^{2} )^{2} \)
13$C_2$ \( ( 1 + 242 T + p^{5} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2738338 T^{2} + p^{10} T^{4} \)
19$C_2^2$ \( 1 - 3380474 T^{2} + p^{10} T^{4} \)
23$C_2$ \( ( 1 - 1296 T + p^{5} T^{2} )^{2} \)
29$C_2^2$ \( 1 - 37062298 T^{2} + p^{10} T^{4} \)
31$C_2^2$ \( 1 - 45678866 T^{2} + p^{10} T^{4} \)
37$C_2$ \( ( 1 + 12058 T + p^{5} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 12512146 T^{2} + p^{10} T^{4} \)
43$C_2^2$ \( 1 + 33190390 T^{2} + p^{10} T^{4} \)
47$C_2$ \( ( 1 - 12960 T + p^{5} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 108251530 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 - 8424 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 25762 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 2596035290 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 55728 T + p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 26026 T + p^{5} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 5786874674 T^{2} + p^{10} T^{4} \)
83$C_2$ \( ( 1 + 78408 T + p^{5} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 4075828594 T^{2} + p^{10} T^{4} \)
97$C_2$ \( ( 1 - 103090 T + p^{5} T^{2} )^{2} \)
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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

−14.71129369044257939475333131646, −14.36317740100824711612657758064, −13.96887580908310312042156693761, −13.57885586685863235963073695363, −12.37236898804660655551309058413, −12.32879673888747014444068254005, −11.53693801456430301554361452429, −10.78109304667281320493211053787, −9.755764718281155765990687166326, −9.434335189512990665071264866995, −8.669619529935670120078334942163, −8.665040761004924492090964214046, −7.15630205632918575835927423397, −7.13607741697556791976975282665, −6.15057481462694374652141959374, −4.82021104028938605631819022182, −3.84593347347276326113220606827, −3.42020354654293110599545564520, −2.10416020590992095632850889787, −1.19374229243361420945347647272, 1.19374229243361420945347647272, 2.10416020590992095632850889787, 3.42020354654293110599545564520, 3.84593347347276326113220606827, 4.82021104028938605631819022182, 6.15057481462694374652141959374, 7.13607741697556791976975282665, 7.15630205632918575835927423397, 8.665040761004924492090964214046, 8.669619529935670120078334942163, 9.434335189512990665071264866995, 9.755764718281155765990687166326, 10.78109304667281320493211053787, 11.53693801456430301554361452429, 12.32879673888747014444068254005, 12.37236898804660655551309058413, 13.57885586685863235963073695363, 13.96887580908310312042156693761, 14.36317740100824711612657758064, 14.71129369044257939475333131646

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