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Properties

Label 4-3e10-1.1-c1e2-0-3
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

Related objects

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·4-s + 10·7-s + 4·13-s + 12·16-s + 16·19-s − 10·25-s − 40·28-s − 14·31-s − 2·37-s − 26·43-s + 61·49-s − 16·52-s − 2·61-s − 32·64-s + 10·67-s − 14·73-s − 64·76-s − 8·79-s + 40·91-s + 28·97-s + 40·100-s − 26·103-s + 34·109-s + 120·112-s − 22·121-s + 56·124-s + 127-s + ⋯
L(s)  = 1  − 2·4-s + 3.77·7-s + 1.10·13-s + 3·16-s + 3.67·19-s − 2·25-s − 7.55·28-s − 2.51·31-s − 0.328·37-s − 3.96·43-s + 61/7·49-s − 2.21·52-s − 0.256·61-s − 4·64-s + 1.22·67-s − 1.63·73-s − 7.34·76-s − 0.900·79-s + 4.19·91-s + 2.84·97-s + 4·100-s − 2.56·103-s + 3.25·109-s + 11.3·112-s − 2·121-s + 5.02·124-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: \(0\)
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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.7.ak_bn
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
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$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.31.o_eh
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.37.c_cx
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.43.ba_jv
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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
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$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−14.55465593186520144639783075046, −13.98333447171136912436166514168, −13.98333447171136912436166514168, −13.24978405566496793478436658265, −13.24978405566496793478436658265, −11.91890658630114301782409186133, −11.91890658630114301782409186133, −11.30223235079307772334680977560, −11.30223235079307772334680977560, −10.15270656474832309983635908637, −10.15270656474832309983635908637, −9.086003831873135423732654139431, −9.086003831873135423732654139431, −8.203092356009887426190725310168, −8.203092356009887426190725310168, −7.47986805287089464979148940141, −7.47986805287089464979148940141, −5.57920885013260782198005848714, −5.57920885013260782198005848714, −4.89221274974875080133813941614, −4.89221274974875080133813941614, −3.68456101574470316351634814116, −3.68456101574470316351634814116, −1.47816197657666824979335991741, −1.47816197657666824979335991741, 1.47816197657666824979335991741, 1.47816197657666824979335991741, 3.68456101574470316351634814116, 3.68456101574470316351634814116, 4.89221274974875080133813941614, 4.89221274974875080133813941614, 5.57920885013260782198005848714, 5.57920885013260782198005848714, 7.47986805287089464979148940141, 7.47986805287089464979148940141, 8.203092356009887426190725310168, 8.203092356009887426190725310168, 9.086003831873135423732654139431, 9.086003831873135423732654139431, 10.15270656474832309983635908637, 10.15270656474832309983635908637, 11.30223235079307772334680977560, 11.30223235079307772334680977560, 11.91890658630114301782409186133, 11.91890658630114301782409186133, 13.24978405566496793478436658265, 13.24978405566496793478436658265, 13.98333447171136912436166514168, 13.98333447171136912436166514168, 14.55465593186520144639783075046

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