Properties

Label 4-40e4-1.1-c1e2-0-34
Degree $4$
Conductor $2560000$
Sign $1$
Analytic cond. $163.227$
Root an. cond. $3.57436$
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  + 2·3-s − 2·7-s + 2·9-s + 8·13-s + 8·17-s + 8·19-s − 4·21-s − 10·23-s + 6·27-s + 16·39-s − 8·41-s + 14·43-s − 6·47-s + 2·49-s + 16·51-s + 8·53-s + 16·57-s + 8·59-s + 16·61-s − 4·63-s − 6·67-s − 20·69-s − 8·73-s + 16·79-s + 11·81-s − 10·83-s − 16·91-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 2/3·9-s + 2.21·13-s + 1.94·17-s + 1.83·19-s − 0.872·21-s − 2.08·23-s + 1.15·27-s + 2.56·39-s − 1.24·41-s + 2.13·43-s − 0.875·47-s + 2/7·49-s + 2.24·51-s + 1.09·53-s + 2.11·57-s + 1.04·59-s + 2.04·61-s − 0.503·63-s − 0.733·67-s − 2.40·69-s − 0.936·73-s + 1.80·79-s + 11/9·81-s − 1.09·83-s − 1.67·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2560000\)    =    \(2^{12} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(163.227\)
Root analytic conductor: \(3.57436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2560000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.560200360\)
\(L(\frac12)\) \(\approx\) \(4.560200360\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_c
7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_c
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.13.ai_bg
17$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.17.ai_bg
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.23.k_by
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.29.a_acc
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 + p^{2} T^{4} \) 2.37.a_a
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.41.i_du
43$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.43.ao_du
47$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_s
53$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_bg
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.61.aq_he
67$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.67.g_s
71$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.71.a_ek
73$C_2^2$ \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_bg
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.83.k_by
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.a_ada
97$C_2^2$ \( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.97.y_lc
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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

−9.753359233865044264880490250687, −9.233876344920512687445257134899, −8.621693339227168984376396147895, −8.377611628205301960469594209968, −8.251612302347543166735109945931, −7.71939737862736182821158411714, −7.18634752015541410659950370014, −7.03970656593713157818804602142, −6.21033591362749536666307052049, −6.05276863846924168607004081979, −5.48885816623283446822578987318, −5.33930526209935808973419937265, −4.33954070422947461421297813389, −3.90847410744804246205584291968, −3.51348826478406139020965012961, −3.29627641145198000554898142601, −2.78633052877549555790708846992, −2.09798820025580161533814140408, −1.28377457687275747464230304184, −0.902555752359531069755693787071, 0.902555752359531069755693787071, 1.28377457687275747464230304184, 2.09798820025580161533814140408, 2.78633052877549555790708846992, 3.29627641145198000554898142601, 3.51348826478406139020965012961, 3.90847410744804246205584291968, 4.33954070422947461421297813389, 5.33930526209935808973419937265, 5.48885816623283446822578987318, 6.05276863846924168607004081979, 6.21033591362749536666307052049, 7.03970656593713157818804602142, 7.18634752015541410659950370014, 7.71939737862736182821158411714, 8.251612302347543166735109945931, 8.377611628205301960469594209968, 8.621693339227168984376396147895, 9.233876344920512687445257134899, 9.753359233865044264880490250687

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