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Properties

Label 4-3e10-1.1-c1e2-0-1
Degree $4$
Conductor $59049$
Sign $1$
Analytic cond. $3.76501$
Root an. cond. $1.39296$
Motivic weight $1$
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  + 2·4-s − 5·7-s − 2·13-s + 16·19-s + 5·25-s − 10·28-s + 7·31-s − 2·37-s + 13·43-s + 7·49-s − 4·52-s + 61-s − 8·64-s − 5·67-s − 14·73-s + 32·76-s + 4·79-s + 10·91-s − 14·97-s + 10·100-s + 13·103-s + 34·109-s + 11·121-s + 14·124-s + 127-s + 131-s − 80·133-s + ⋯
L(s)  = 1  + 4-s − 1.88·7-s − 0.554·13-s + 3.67·19-s + 25-s − 1.88·28-s + 1.25·31-s − 0.328·37-s + 1.98·43-s + 49-s − 0.554·52-s + 0.128·61-s − 64-s − 0.610·67-s − 1.63·73-s + 3.67·76-s + 0.450·79-s + 1.04·91-s − 1.42·97-s + 100-s + 1.28·103-s + 3.25·109-s + 121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s − 6.93·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(59049\)    =    \(3^{10}\)
Sign: $1$
Analytic conductor: \(3.76501\)
Root analytic conductor: \(1.39296\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 59049,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.500426299\)
\(L(\frac12)\) \(\approx\) \(1.500426299\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
good2$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.2.a_ac
5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
7$C_2$ \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.f_s
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.c_aj
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.19.aq_dy
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.29.a_abd
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.ah_s
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.43.an_ew
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.61.ab_aci
67$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.f_abq
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.ae_acl
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) 2.97.o_dv
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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

−12.17216084577178593442409839152, −11.80479450727914061172521559686, −11.66444610340829028676463276094, −10.88929639506203710607731869380, −10.38852928418118590507409540242, −9.889773234492330283987980611785, −9.479731640275515217364893870697, −9.269336445200337117395732519387, −8.493594547407073719153542348265, −7.53890171549350114623717657307, −7.34999927131253174743312079833, −6.95969124298714703379155794014, −6.29299066977873031604210086520, −5.88438083687415484646604778243, −5.23265169556691067257601445197, −4.50769642737245794439515982991, −3.34897062892099219476769787929, −3.10431537897668476110456106375, −2.54234440288950074036639420280, −1.05430520430552585684492675324, 1.05430520430552585684492675324, 2.54234440288950074036639420280, 3.10431537897668476110456106375, 3.34897062892099219476769787929, 4.50769642737245794439515982991, 5.23265169556691067257601445197, 5.88438083687415484646604778243, 6.29299066977873031604210086520, 6.95969124298714703379155794014, 7.34999927131253174743312079833, 7.53890171549350114623717657307, 8.493594547407073719153542348265, 9.269336445200337117395732519387, 9.479731640275515217364893870697, 9.889773234492330283987980611785, 10.38852928418118590507409540242, 10.88929639506203710607731869380, 11.66444610340829028676463276094, 11.80479450727914061172521559686, 12.17216084577178593442409839152

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