Properties

Label 4-60e4-1.1-c1e2-0-28
Degree $4$
Conductor $12960000$
Sign $1$
Analytic cond. $826.340$
Root an. cond. $5.36154$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·19-s − 16·31-s − 14·49-s + 4·61-s − 32·79-s − 28·109-s − 22·121-s + ⋯
L(s)  = 1  − 1.83·19-s − 2.87·31-s − 2·49-s + 0.512·61-s − 3.60·79-s − 2.68·109-s − 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(826.340\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 12960000,\ (\ :1/2, 1/2),\ 1)\)

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 $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.17.a_o
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.23.a_abi
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.37.a_cw
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.43.a_di
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.47.a_o
53$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.53.a_di
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.67.a_fe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.73.a_fq
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \) 2.83.a_afy
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.97.a_hm
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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

−8.366929638720068752681651976524, −8.059492953441077715721486581214, −7.61201764492633572828264296859, −7.26465322006153058276021401323, −6.82390765211343474306049983915, −6.64110706226956238394306696091, −5.97577321334214654654774648756, −5.91829988567615582973787674166, −5.26066871631210117691861364451, −5.07045094425996385877981998071, −4.40432398021246948216256600780, −4.19477066569243289082313729622, −3.58919177503697992681102310607, −3.44666720488530537383451545791, −2.51600714639245015337437518452, −2.45602772843344046098083229385, −1.51742371842981158249503352735, −1.49731763651313745996414657237, 0, 0, 1.49731763651313745996414657237, 1.51742371842981158249503352735, 2.45602772843344046098083229385, 2.51600714639245015337437518452, 3.44666720488530537383451545791, 3.58919177503697992681102310607, 4.19477066569243289082313729622, 4.40432398021246948216256600780, 5.07045094425996385877981998071, 5.26066871631210117691861364451, 5.91829988567615582973787674166, 5.97577321334214654654774648756, 6.64110706226956238394306696091, 6.82390765211343474306049983915, 7.26465322006153058276021401323, 7.61201764492633572828264296859, 8.059492953441077715721486581214, 8.366929638720068752681651976524

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