Properties

Label 4-208e2-1.1-c5e2-0-6
Degree $4$
Conductor $43264$
Sign $1$
Analytic cond. $1112.87$
Root an. cond. $5.77579$
Motivic weight $5$
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  + 26·3-s + 21·9-s − 1.19e3·13-s − 2.20e3·17-s − 2.10e3·23-s + 3.64e3·25-s − 1.01e4·27-s − 8.20e3·29-s − 3.10e4·39-s − 1.99e4·43-s + 2.25e4·49-s − 5.72e4·51-s − 1.50e3·53-s − 1.15e5·61-s − 5.46e4·69-s + 9.48e4·75-s − 1.26e5·79-s − 1.72e5·81-s − 2.13e5·87-s − 2.26e5·101-s − 5.00e4·103-s − 4.98e4·107-s + 2.00e5·113-s − 2.51e4·117-s + 3.07e5·121-s + 127-s − 5.19e5·129-s + ⋯
L(s)  = 1  + 1.66·3-s + 7/81·9-s − 1.96·13-s − 1.84·17-s − 0.827·23-s + 1.16·25-s − 2.68·27-s − 1.81·29-s − 3.27·39-s − 1.64·43-s + 1.34·49-s − 3.08·51-s − 0.0733·53-s − 3.98·61-s − 1.38·69-s + 1.94·75-s − 2.27·79-s − 2.92·81-s − 3.02·87-s − 2.20·101-s − 0.465·103-s − 0.420·107-s + 1.47·113-s − 0.169·117-s + 1.91·121-s + 5.50e−6·127-s − 2.74·129-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(43264\)    =    \(2^{8} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1112.87\)
Root analytic conductor: \(5.77579\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 43264,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
13$C_2$ \( 1 + 92 p T + p^{5} T^{2} \)
good3$C_2$ \( ( 1 - 13 T + p^{5} T^{2} )^{2} \)
5$C_2^2$ \( 1 - 3649 T^{2} + p^{10} T^{4} \)
7$C_2^2$ \( 1 - 461 p^{2} T^{2} + p^{10} T^{4} \)
11$C_2^2$ \( 1 - 307702 T^{2} + p^{10} T^{4} \)
17$C_2$ \( ( 1 + 1101 T + p^{5} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 3583298 T^{2} + p^{10} T^{4} \)
23$C_2$ \( ( 1 + 1050 T + p^{5} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 4104 T + p^{5} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 35363074 T^{2} + p^{10} T^{4} \)
37$C_2^2$ \( 1 - 62841233 T^{2} + p^{10} T^{4} \)
41$C_2^2$ \( 1 - 141842002 T^{2} + p^{10} T^{4} \)
43$C_2$ \( ( 1 + 9995 T + p^{5} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 450028765 T^{2} + p^{10} T^{4} \)
53$C_2$ \( ( 1 + 750 T + p^{5} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 246071246 T^{2} + p^{10} T^{4} \)
61$C_2$ \( ( 1 + 57920 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 2179862870 T^{2} + p^{10} T^{4} \)
71$C_2^2$ \( 1 + 454456379 T^{2} + p^{10} T^{4} \)
73$C_2^2$ \( 1 - 680937230 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 63202 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 4802491522 T^{2} + p^{10} T^{4} \)
89$C_2^2$ \( 1 - 189689614 T^{2} + p^{10} T^{4} \)
97$C_2^2$ \( 1 + 8570163790 T^{2} + p^{10} T^{4} \)
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

−11.07518505427642931373588017154, −10.95845871089978335347092100929, −10.07240061383184630196794295625, −9.613437076527661460268641206209, −9.086674368057515208421865010271, −8.935208404243323166050664514314, −8.281876119752454822123031392567, −7.903525462519486332437382595860, −7.16193026398581799201107308011, −7.04041124850421519035628995697, −5.97710239461086149698138786710, −5.52351941941242903267655947756, −4.58840901055223369668989596583, −4.28231856942894561607070965770, −3.17317319576574243793865835401, −2.97835122906630426785308206674, −2.10018479302355757072476808650, −1.91835734953298521873449231735, 0, 0, 1.91835734953298521873449231735, 2.10018479302355757072476808650, 2.97835122906630426785308206674, 3.17317319576574243793865835401, 4.28231856942894561607070965770, 4.58840901055223369668989596583, 5.52351941941242903267655947756, 5.97710239461086149698138786710, 7.04041124850421519035628995697, 7.16193026398581799201107308011, 7.903525462519486332437382595860, 8.281876119752454822123031392567, 8.935208404243323166050664514314, 9.086674368057515208421865010271, 9.613437076527661460268641206209, 10.07240061383184630196794295625, 10.95845871089978335347092100929, 11.07518505427642931373588017154

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