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Properties

Label 4-960e2-1.1-c1e2-0-16
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
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 + 2·5-s − 6·7-s + 3·9-s + 2·11-s + 4·15-s + 2·17-s + 2·19-s − 12·21-s − 10·23-s − 25-s + 4·27-s − 10·29-s + 4·33-s − 12·35-s + 6·45-s + 6·47-s + 18·49-s + 4·51-s + 12·53-s + 4·55-s + 4·57-s − 10·59-s − 10·61-s − 18·63-s − 20·69-s + 32·71-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s − 2.26·7-s + 9-s + 0.603·11-s + 1.03·15-s + 0.485·17-s + 0.458·19-s − 2.61·21-s − 2.08·23-s − 1/5·25-s + 0.769·27-s − 1.85·29-s + 0.696·33-s − 2.02·35-s + 0.894·45-s + 0.875·47-s + 18/7·49-s + 0.560·51-s + 1.64·53-s + 0.539·55-s + 0.529·57-s − 1.30·59-s − 1.28·61-s − 2.26·63-s − 2.40·69-s + 3.79·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.585156365\)
\(L(\frac12)\) \(\approx\) \(2.585156365\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.7.g_s
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.11.ac_c
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_c
19$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_c
23$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.23.k_by
29$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_by
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.31.a_aba
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.37.a_acg
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2^2$ \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.47.ag_s
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.59.k_by
61$C_2^2$ \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_by
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.71.abg_pi
73$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.73.as_gg
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.83.i_ha
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_c
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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

−10.06380593436951729628552068151, −9.596604652362372587029276120305, −9.475812637329919046617796510225, −9.229320557790703480107608635861, −8.809182827774434211050864656263, −8.054119616210688932506063871974, −7.75396279651640104039603863617, −7.39747542274084604520690074712, −6.71102150224711344248417955574, −6.31881093001393539583585353049, −6.22012313833495938743976535894, −5.52838040023866753632566420717, −5.16895156674008812555779876914, −4.09889304521934073595707375657, −3.64641759743484616565197877150, −3.63993969524858877075223870528, −2.90790317134099167171652753252, −2.12087486332222650588068712656, −1.99279708649659521315490156290, −0.67714784877951478815740771660, 0.67714784877951478815740771660, 1.99279708649659521315490156290, 2.12087486332222650588068712656, 2.90790317134099167171652753252, 3.63993969524858877075223870528, 3.64641759743484616565197877150, 4.09889304521934073595707375657, 5.16895156674008812555779876914, 5.52838040023866753632566420717, 6.22012313833495938743976535894, 6.31881093001393539583585353049, 6.71102150224711344248417955574, 7.39747542274084604520690074712, 7.75396279651640104039603863617, 8.054119616210688932506063871974, 8.809182827774434211050864656263, 9.229320557790703480107608635861, 9.475812637329919046617796510225, 9.596604652362372587029276120305, 10.06380593436951729628552068151

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