Properties

Label 4-528e2-1.1-c3e2-0-6
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 $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s − 6·5-s − 10·7-s + 27·9-s − 22·11-s − 2·13-s − 36·15-s − 104·17-s − 4·19-s − 60·21-s + 102·23-s − 206·25-s + 108·27-s − 392·29-s + 64·31-s − 132·33-s + 60·35-s − 164·37-s − 12·39-s − 732·41-s − 168·43-s − 162·45-s + 314·47-s − 594·49-s − 624·51-s − 382·53-s + 132·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.536·5-s − 0.539·7-s + 9-s − 0.603·11-s − 0.0426·13-s − 0.619·15-s − 1.48·17-s − 0.0482·19-s − 0.623·21-s + 0.924·23-s − 1.64·25-s + 0.769·27-s − 2.51·29-s + 0.370·31-s − 0.696·33-s + 0.289·35-s − 0.728·37-s − 0.0492·39-s − 2.78·41-s − 0.595·43-s − 0.536·45-s + 0.974·47-s − 1.73·49-s − 1.71·51-s − 0.990·53-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: \(2\)
Selberg data: \((4,\ 278784,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$D_{4}$ \( 1 + 6 T + 242 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 10 T + 694 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 2 T - 518 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 104 T + 7022 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 4 T - 1578 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 102 T + 24062 T^{2} - 102 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 392 T + 75702 T^{2} + 392 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 64 T + 33406 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 164 T - 770 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 732 T + 264998 T^{2} + 732 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 168 T + 2034 p T^{2} + 168 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 314 T + 4210 p T^{2} - 314 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 382 T + 310962 T^{2} + 382 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 508 T + 418086 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 66354 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 216 T + 298758 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 878 T + 841070 T^{2} - 878 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 260 T - 58058 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 118 T + 829606 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 496 T + 441030 T^{2} + 496 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 1756 T + 1701014 T^{2} + 1756 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 1968 T + 2769054 T^{2} - 1968 p^{3} T^{3} + p^{6} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.912405195793462309920735783128, −9.860855441564568517159637555797, −9.231084084779624122174398597821, −8.958896803754636278908215627123, −8.346233653560143744736904519605, −8.125075571621738972200171035291, −7.44068169875648795437332552054, −7.31200563800412717302502943301, −6.49017230289101919910898183570, −6.40491472909925589863170417806, −5.23323335407191536790136301404, −5.18315595222357040805036811438, −4.20051098005224296151715928499, −3.90808015641447671499410057944, −3.25286253769863108259254078661, −2.89979216271398834782547103014, −1.90350896958212712454798310565, −1.73584993132922050774471283261, 0, 0, 1.73584993132922050774471283261, 1.90350896958212712454798310565, 2.89979216271398834782547103014, 3.25286253769863108259254078661, 3.90808015641447671499410057944, 4.20051098005224296151715928499, 5.18315595222357040805036811438, 5.23323335407191536790136301404, 6.40491472909925589863170417806, 6.49017230289101919910898183570, 7.31200563800412717302502943301, 7.44068169875648795437332552054, 8.125075571621738972200171035291, 8.346233653560143744736904519605, 8.958896803754636278908215627123, 9.231084084779624122174398597821, 9.860855441564568517159637555797, 9.912405195793462309920735783128

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