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Properties

Label 4-3e10-1.1-c1e2-0-5
Degree $4$
Conductor $59049$
Sign $-1$
Analytic cond. $3.76501$
Root an. cond. $1.39296$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 2·7-s + 13-s − 3·16-s − 2·19-s − 7·25-s + 2·28-s − 8·31-s + 7·37-s − 2·43-s + 7·49-s − 52-s − 14·61-s + 7·64-s − 20·67-s − 14·73-s + 2·76-s − 2·79-s − 2·91-s − 2·97-s + 7·100-s + 101-s + 103-s + 107-s + 109-s + 6·112-s + 113-s + ⋯
L(s)  = 1  − 1/2·4-s − 0.755·7-s + 0.277·13-s − 3/4·16-s − 0.458·19-s − 7/5·25-s + 0.377·28-s − 1.43·31-s + 1.15·37-s − 0.304·43-s + 49-s − 0.138·52-s − 1.79·61-s + 7/8·64-s − 2.44·67-s − 1.63·73-s + 0.229·76-s − 0.225·79-s − 0.209·91-s − 0.203·97-s + 7/10·100-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.566·112-s + 0.0940·113-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: yes
Self-dual: yes
Analytic rank: \(1\)
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$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.2.a_b
5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_ad
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) 2.13.ab_am
17$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.17.a_h
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_ap
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.23.a_bi
29$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.29.a_cd
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bh
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
41$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.41.a_bi
43$C_2^2$ \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_abn
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.47.a_bu
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.61.o_gp
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.67.u_ja
71$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.71.a_bi
73$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.73.o_hn
79$C_2^2$ \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.79.c_acx
83$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.83.a_aba
89$C_2^2$ \( 1 + 151 T^{2} + p^{2} T^{4} \) 2.89.a_fv
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.97.c_adp
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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

−14.7834538642, −14.3222157182, −13.7952588753, −13.3867300103, −13.0911595174, −12.7418699243, −11.9758276314, −11.7724145898, −11.1403805066, −10.6132994579, −10.2216602024, −9.60414902741, −9.09651518350, −8.99370561256, −8.21158033833, −7.61443371658, −7.20384099735, −6.48867585133, −5.94967085682, −5.58199868911, −4.58907933275, −4.21660573942, −3.50737832038, −2.72297727784, −1.74070911643, 0, 1.74070911643, 2.72297727784, 3.50737832038, 4.21660573942, 4.58907933275, 5.58199868911, 5.94967085682, 6.48867585133, 7.20384099735, 7.61443371658, 8.21158033833, 8.99370561256, 9.09651518350, 9.60414902741, 10.2216602024, 10.6132994579, 11.1403805066, 11.7724145898, 11.9758276314, 12.7418699243, 13.0911595174, 13.3867300103, 13.7952588753, 14.3222157182, 14.7834538642

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