Properties

Label 4-528e2-1.1-c3e2-0-1
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  + 6·3-s − 6·5-s − 16·7-s + 27·9-s + 22·11-s + 22·13-s − 36·15-s − 22·17-s + 62·19-s − 96·21-s + 210·23-s − 86·25-s + 108·27-s − 214·29-s + 328·31-s + 132·33-s + 96·35-s − 200·37-s + 132·39-s + 114·41-s + 606·43-s − 162·45-s + 382·47-s + 54·49-s − 132·51-s − 2·53-s − 132·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.536·5-s − 0.863·7-s + 9-s + 0.603·11-s + 0.469·13-s − 0.619·15-s − 0.313·17-s + 0.748·19-s − 0.997·21-s + 1.90·23-s − 0.687·25-s + 0.769·27-s − 1.37·29-s + 1.90·31-s + 0.696·33-s + 0.463·35-s − 0.888·37-s + 0.541·39-s + 0.434·41-s + 2.14·43-s − 0.536·45-s + 1.18·47-s + 0.157·49-s − 0.362·51-s − 0.00518·53-s − 0.323·55-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

\(L(2)\) \(\approx\) \(4.299928009\)
\(L(\frac12)\) \(\approx\) \(4.299928009\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
11$C_1$ \( ( 1 - p T )^{2} \)
good5$D_{4}$ \( 1 + 6 T + 122 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 16 T + 202 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 22 T + 4378 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 22 T + 8714 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 62 T + 13446 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 210 T + 31934 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 214 T + 53514 T^{2} + 214 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 328 T + 84286 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 200 T + 110758 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 114 T + 25874 T^{2} - 114 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 606 T + 211230 T^{2} - 606 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 382 T + 183710 T^{2} - 382 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 182538 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 1084 T + 703974 T^{2} - 1084 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 354 T + 70866 T^{2} + 354 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 456 T + 644742 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 34 T + 616238 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 112 T + 109870 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 820 T + 1099378 T^{2} + 820 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1012 T + 478422 T^{2} - 1012 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1036 T + 1658534 T^{2} - 1036 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 1308 T + 2217990 T^{2} - 1308 p^{3} T^{3} + p^{6} T^{4} \)
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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

−10.45529422617606057435322949710, −10.26001177346140582346102935388, −9.562742068572166329068927385394, −9.269753314993883069580836332801, −8.871443552694200212055445436760, −8.688873198532775989955429883153, −7.84078226141835357182688160713, −7.65160424244876174326277286536, −6.97090224287737609991899310803, −6.89534843857303951330593241239, −6.02367274163203450089865376390, −5.71831224866765406969633133236, −4.81927931566984915424509187485, −4.35218576220595740621841208660, −3.59774615519289619573614399993, −3.55101930564652665654918253569, −2.74362186559155906091669894778, −2.27594757165313731794705150311, −1.22036072693743274384715117354, −0.66282684364778518201295839902, 0.66282684364778518201295839902, 1.22036072693743274384715117354, 2.27594757165313731794705150311, 2.74362186559155906091669894778, 3.55101930564652665654918253569, 3.59774615519289619573614399993, 4.35218576220595740621841208660, 4.81927931566984915424509187485, 5.71831224866765406969633133236, 6.02367274163203450089865376390, 6.89534843857303951330593241239, 6.97090224287737609991899310803, 7.65160424244876174326277286536, 7.84078226141835357182688160713, 8.688873198532775989955429883153, 8.871443552694200212055445436760, 9.269753314993883069580836332801, 9.562742068572166329068927385394, 10.26001177346140582346102935388, 10.45529422617606057435322949710

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