↑

Properties

Label 4-484128-1.1-c1e2-0-1
Degree $4$
Conductor $484128$
Sign $1$
Analytic cond. $30.8684$
Root an. cond. $2.35710$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 4·5-s − 8-s + 9-s + 4·10-s + 8·13-s + 16-s − 4·17-s − 18-s − 4·20-s + 2·25-s − 8·26-s − 32-s + 4·34-s + 36-s + 4·37-s + 4·40-s − 2·41-s − 4·45-s − 10·49-s − 2·50-s + 8·52-s − 8·53-s + 20·61-s + 64-s − 32·65-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 2.21·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.894·20-s + 2/5·25-s − 1.56·26-s − 0.176·32-s + 0.685·34-s + 1/6·36-s + 0.657·37-s + 0.632·40-s − 0.312·41-s − 0.596·45-s − 1.42·49-s − 0.282·50-s + 1.10·52-s − 1.09·53-s + 2.56·61-s + 1/8·64-s − 3.96·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 484128 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484128 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(484128\)    =    \(2^{5} \cdot 3^{2} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(30.8684\)
Root analytic conductor: \(2.35710\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 484128,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7782511827\)
\(L(\frac12)\) \(\approx\) \(0.7782511827\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 + T \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
41$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.43.a_acg
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.53.i_es
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.a_bq
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.79.a_abm
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 + 18 T + p T^{2} )^{2} \) 2.97.bk_ty
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

−8.401315341994798884355247872006, −8.206997423160705401586467084746, −7.70369193842423959851529250195, −7.38061286309981987798076407807, −6.75149197996895509149400331342, −6.34610400633838841606709758740, −6.01525457350817680510353362862, −5.23259258555002052173439710044, −4.50163197172141360084956861046, −4.12781062486135278830601208813, −3.51127392821665200397122615358, −3.37335197437025672177366782515, −2.26882612979336872993578884896, −1.47945276536207297298646495855, −0.55423486549685232634584617452, 0.55423486549685232634584617452, 1.47945276536207297298646495855, 2.26882612979336872993578884896, 3.37335197437025672177366782515, 3.51127392821665200397122615358, 4.12781062486135278830601208813, 4.50163197172141360084956861046, 5.23259258555002052173439710044, 6.01525457350817680510353362862, 6.34610400633838841606709758740, 6.75149197996895509149400331342, 7.38061286309981987798076407807, 7.70369193842423959851529250195, 8.206997423160705401586467084746, 8.401315341994798884355247872006

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