Properties

Label 4-528e2-1.1-c3e2-0-2
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·3-s + 37·9-s + 74·25-s + 80·27-s + 680·31-s + 868·37-s + 686·49-s − 832·67-s + 592·75-s − 359·81-s + 5.44e3·93-s − 68·97-s − 2.34e3·103-s + 6.94e3·111-s − 1.33e3·121-s + 127-s + 131-s + 137-s + 139-s + 5.48e3·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + 173-s + ⋯
L(s)  = 1  + 1.53·3-s + 1.37·9-s + 0.591·25-s + 0.570·27-s + 3.93·31-s + 3.85·37-s + 2·49-s − 1.51·67-s + 0.911·75-s − 0.492·81-s + 6.06·93-s − 0.0711·97-s − 2.24·103-s + 5.93·111-s − 121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 3.07·147-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2·169-s + 0.000439·173-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: \(0\)
Selberg data: \((4,\ 278784,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.556784289\)
\(L(\frac12)\) \(\approx\) \(6.556784289\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - 8 T + p^{3} T^{2} \)
11$C_2$ \( 1 + p^{3} T^{2} \)
good5$C_2$ \( ( 1 - 18 T + p^{3} T^{2} )( 1 + 18 T + p^{3} T^{2} ) \)
7$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
13$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
17$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
23$C_2$ \( ( 1 - 108 T + p^{3} T^{2} )( 1 + 108 T + p^{3} T^{2} ) \)
29$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 340 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 434 T + p^{3} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
47$C_2$ \( ( 1 - 36 T + p^{3} T^{2} )( 1 + 36 T + p^{3} T^{2} ) \)
53$C_2$ \( ( 1 - 738 T + p^{3} T^{2} )( 1 + 738 T + p^{3} T^{2} ) \)
59$C_2$ \( ( 1 - 720 T + p^{3} T^{2} )( 1 + 720 T + p^{3} T^{2} ) \)
61$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 416 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 612 T + p^{3} T^{2} )( 1 + 612 T + p^{3} T^{2} ) \)
73$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 - 1674 T + p^{3} T^{2} )( 1 + 1674 T + p^{3} T^{2} ) \)
97$C_2$ \( ( 1 + 34 T + p^{3} T^{2} )^{2} \)
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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

−10.37613924648900113921443241424, −10.20275314904884750559653426040, −9.631767891256481276215227649560, −9.363688946054828883793276745602, −8.869085305406370103034040455311, −8.353002953905309692097089911773, −8.071240949668161094237269354166, −7.71868115489405103041697559618, −7.21439585164361916139636571434, −6.54020701606979949016270937395, −6.20567281514135934671935817724, −5.63729548565605068704451694074, −4.71351016195649338200593897682, −4.36287794177338476391793176566, −4.01461285889422637827228481262, −3.07493383302392991257554735947, −2.58083184588445317408101690410, −2.52206896754509317923614730574, −1.26318378881351961847256467860, −0.798545620183460543886279979259, 0.798545620183460543886279979259, 1.26318378881351961847256467860, 2.52206896754509317923614730574, 2.58083184588445317408101690410, 3.07493383302392991257554735947, 4.01461285889422637827228481262, 4.36287794177338476391793176566, 4.71351016195649338200593897682, 5.63729548565605068704451694074, 6.20567281514135934671935817724, 6.54020701606979949016270937395, 7.21439585164361916139636571434, 7.71868115489405103041697559618, 8.071240949668161094237269354166, 8.353002953905309692097089911773, 8.869085305406370103034040455311, 9.363688946054828883793276745602, 9.631767891256481276215227649560, 10.20275314904884750559653426040, 10.37613924648900113921443241424

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