↑

Properties

Label 4-3024e2-1.1-c1e2-0-32
Degree $4$
Conductor $9144576$
Sign $1$
Analytic cond. $583.066$
Root an. cond. $4.91393$
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  + 2·5-s + 7-s + 3·11-s + 6·13-s − 14·17-s − 2·19-s + 4·23-s + 5·25-s − 4·29-s + 10·31-s + 2·35-s − 12·37-s + 7·41-s + 11·43-s − 8·47-s + 8·53-s + 6·55-s − 9·59-s − 4·61-s + 12·65-s + 9·67-s + 26·73-s + 3·77-s + 10·79-s − 28·85-s − 20·89-s + 6·91-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.377·7-s + 0.904·11-s + 1.66·13-s − 3.39·17-s − 0.458·19-s + 0.834·23-s + 25-s − 0.742·29-s + 1.79·31-s + 0.338·35-s − 1.97·37-s + 1.09·41-s + 1.67·43-s − 1.16·47-s + 1.09·53-s + 0.809·55-s − 1.17·59-s − 0.512·61-s + 1.48·65-s + 1.09·67-s + 3.04·73-s + 0.341·77-s + 1.12·79-s − 3.03·85-s − 2.11·89-s + 0.628·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9144576\)    =    \(2^{8} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(583.066\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9144576,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.540960728\)
\(L(\frac12)\) \(\approx\) \(3.540960728\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2^2$ \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.13.ag_x
17$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.17.o_df
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_ah
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_an
31$C_2^2$ \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.31.ak_cr
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.37.m_eg
41$C_2^2$ \( 1 - 7 T + 8 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.41.ah_i
43$C_2^2$ \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.43.al_da
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_r
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2^2$ \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.59.j_w
61$C_2^2$ \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.61.e_abt
67$C_2^2$ \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.67.aj_o
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.73.aba_md
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_v
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.89.u_ks
97$C_2^2$ \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.97.h_abw
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

−8.908293110457171107099946544314, −8.556445886798652543988696304271, −8.336712215107773586376515877181, −8.033888584706591706501465227680, −7.05008913603601239231417484614, −6.98023831082236461962561096866, −6.67389264911134285087102454765, −6.19711627725221995093528450979, −6.12950406590261967610152802013, −5.50782758879027996837679472859, −4.97867657599210088956583438386, −4.66196834961513079603373925205, −4.11992831882643716261625713497, −4.00387616593165110208195189123, −3.33514538041801455050910422042, −2.68873000176216713956614496240, −2.25866294302774042266442266518, −1.83607417044362600874233249935, −1.29535007741208165161316550300, −0.61249580301976883138315263209, 0.61249580301976883138315263209, 1.29535007741208165161316550300, 1.83607417044362600874233249935, 2.25866294302774042266442266518, 2.68873000176216713956614496240, 3.33514538041801455050910422042, 4.00387616593165110208195189123, 4.11992831882643716261625713497, 4.66196834961513079603373925205, 4.97867657599210088956583438386, 5.50782758879027996837679472859, 6.12950406590261967610152802013, 6.19711627725221995093528450979, 6.67389264911134285087102454765, 6.98023831082236461962561096866, 7.05008913603601239231417484614, 8.033888584706591706501465227680, 8.336712215107773586376515877181, 8.556445886798652543988696304271, 8.908293110457171107099946544314

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