Properties

Label 4-528e2-1.1-c2e2-0-4
Degree $4$
Conductor $278784$
Sign $1$
Analytic cond. $206.984$
Root an. cond. $3.79301$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 16·7-s + 27·9-s + 8·13-s + 12·19-s − 96·21-s + 6·25-s − 108·27-s + 52·31-s + 60·37-s − 48·39-s − 84·43-s + 94·49-s − 72·57-s + 24·61-s + 432·63-s − 4·67-s − 148·73-s − 36·75-s + 80·79-s + 405·81-s + 128·91-s − 312·93-s + 124·97-s − 148·103-s − 400·109-s − 360·111-s + ⋯
L(s)  = 1  − 2·3-s + 16/7·7-s + 3·9-s + 8/13·13-s + 0.631·19-s − 4.57·21-s + 6/25·25-s − 4·27-s + 1.67·31-s + 1.62·37-s − 1.23·39-s − 1.95·43-s + 1.91·49-s − 1.26·57-s + 0.393·61-s + 48/7·63-s − 0.0597·67-s − 2.02·73-s − 0.479·75-s + 1.01·79-s + 5·81-s + 1.40·91-s − 3.35·93-s + 1.27·97-s − 1.43·103-s − 3.66·109-s − 3.24·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(206.984\)
Root analytic conductor: \(3.79301\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 278784,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.868729194\)
\(L(\frac12)\) \(\approx\) \(1.868729194\)
\(L(2)\) 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$C_1$ \( ( 1 + p T )^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 - 6 T^{2} + p^{4} T^{4} \)
7$C_2$ \( ( 1 - 8 T + p^{2} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 402 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 1014 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 98 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 30 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3186 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 42 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 3018 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 2054 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 2562 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 6518 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 74 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 40 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 12194 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 1586 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 - 62 T + p^{2} T^{2} )^{2} \)
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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

−10.92620692475161324516011631602, −10.51810821558973494417326180412, −10.33943251028426868990002767479, −9.628932645442603423512893619920, −9.286202169935506125068647791610, −8.429260650455487585054064908035, −7.990771306290242187444338538061, −7.85515545791436087353982109763, −7.16560744760734825664589802660, −6.56829085631142286162290215295, −6.33835001495353085865767011517, −5.55876087665585514855833515599, −5.27938396910351851496202005640, −4.85540258468475393895855049036, −4.36622202926411974006768008516, −4.05142803184477132938254993222, −2.90725838550722139861647276043, −1.75239054810510256747324559388, −1.38267461020345039610994042751, −0.69800252997721015209033229666, 0.69800252997721015209033229666, 1.38267461020345039610994042751, 1.75239054810510256747324559388, 2.90725838550722139861647276043, 4.05142803184477132938254993222, 4.36622202926411974006768008516, 4.85540258468475393895855049036, 5.27938396910351851496202005640, 5.55876087665585514855833515599, 6.33835001495353085865767011517, 6.56829085631142286162290215295, 7.16560744760734825664589802660, 7.85515545791436087353982109763, 7.990771306290242187444338538061, 8.429260650455487585054064908035, 9.286202169935506125068647791610, 9.628932645442603423512893619920, 10.33943251028426868990002767479, 10.51810821558973494417326180412, 10.92620692475161324516011631602

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