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Properties

Label 4-3e10-1.1-c1e2-0-6
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 $2$

Origins

Related objects

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s − 8·7-s − 14·13-s + 12·16-s − 2·19-s − 10·25-s + 32·28-s + 22·31-s − 20·37-s + 10·43-s + 34·49-s + 56·52-s − 2·61-s − 32·64-s + 10·67-s − 14·73-s + 8·76-s − 26·79-s + 112·91-s + 10·97-s + 40·100-s − 26·103-s − 38·109-s − 96·112-s − 22·121-s − 88·124-s + 127-s + ⋯
L(s)  = 1  − 2·4-s − 3.02·7-s − 3.88·13-s + 3·16-s − 0.458·19-s − 2·25-s + 6.04·28-s + 3.95·31-s − 3.28·37-s + 1.52·43-s + 34/7·49-s + 7.76·52-s − 0.256·61-s − 4·64-s + 1.22·67-s − 1.63·73-s + 0.917·76-s − 2.92·79-s + 11.7·91-s + 1.01·97-s + 4·100-s − 2.56·103-s − 3.63·109-s − 9.07·112-s − 2·121-s − 7.90·124-s + 0.0887·127-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: \(2\)
Selberg data: \((4,\ 59049,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.13.o_cx
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.31.aw_hb
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.37.u_gs
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.43.ak_eh
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.67.ak_gd
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$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.79.ba_mp
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.97.ak_il
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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

−15.22192444218867595431968553628, −14.19891556341339783955983582324, −14.19891556341339783955983582324, −13.45109347493255848216966653118, −13.45109347493255848216966653118, −12.56043437812580641366645635573, −12.56043437812580641366645635573, −12.00563141299999196241656264872, −12.00563141299999196241656264872, −10.16325612920902098193571359997, −10.16325612920902098193571359997, −9.839140997218714811663358114347, −9.839140997218714811663358114347, −8.910401959588073671653022038986, −8.910401959588073671653022038986, −7.67478550580751206163059313011, −7.67478550580751206163059313011, −6.55678077622873572890302306211, −6.55678077622873572890302306211, −5.32179199644498256876875394013, −5.32179199644498256876875394013, −4.15706479427404740125135154289, −4.15706479427404740125135154289, −2.83018486415551887293164017672, −2.83018486415551887293164017672, 0, 0, 2.83018486415551887293164017672, 2.83018486415551887293164017672, 4.15706479427404740125135154289, 4.15706479427404740125135154289, 5.32179199644498256876875394013, 5.32179199644498256876875394013, 6.55678077622873572890302306211, 6.55678077622873572890302306211, 7.67478550580751206163059313011, 7.67478550580751206163059313011, 8.910401959588073671653022038986, 8.910401959588073671653022038986, 9.839140997218714811663358114347, 9.839140997218714811663358114347, 10.16325612920902098193571359997, 10.16325612920902098193571359997, 12.00563141299999196241656264872, 12.00563141299999196241656264872, 12.56043437812580641366645635573, 12.56043437812580641366645635573, 13.45109347493255848216966653118, 13.45109347493255848216966653118, 14.19891556341339783955983582324, 14.19891556341339783955983582324, 15.22192444218867595431968553628

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