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Properties

Label 4-9600e2-1.1-c1e2-0-35
Degree $4$
Conductor $92160000$
Sign $1$
Analytic cond. $5876.20$
Root an. cond. $8.75536$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·7-s + 3·9-s + 4·11-s − 2·13-s + 2·19-s − 4·21-s − 4·23-s + 4·27-s − 6·31-s + 8·33-s − 16·37-s − 4·39-s − 4·41-s − 14·43-s − 12·47-s − 9·49-s + 4·53-s + 4·57-s + 12·59-s − 2·61-s − 6·63-s − 2·67-s − 8·69-s − 8·71-s + 8·73-s − 8·77-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 9-s + 1.20·11-s − 0.554·13-s + 0.458·19-s − 0.872·21-s − 0.834·23-s + 0.769·27-s − 1.07·31-s + 1.39·33-s − 2.63·37-s − 0.640·39-s − 0.624·41-s − 2.13·43-s − 1.75·47-s − 9/7·49-s + 0.549·53-s + 0.529·57-s + 1.56·59-s − 0.256·61-s − 0.755·63-s − 0.244·67-s − 0.963·69-s − 0.949·71-s + 0.936·73-s − 0.911·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(92160000\)    =    \(2^{14} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5876.20\)
Root analytic conductor: \(8.75536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 92160000,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
good7$D_{4}$ \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.7.c_n
11$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_i
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_t
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.17.a_bg
19$D_{4}$ \( 1 - 2 T - 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_al
23$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.23.e_bw
29$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.29.a_i
31$D_{4}$ \( 1 + 6 T + 21 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_v
37$D_{4}$ \( 1 + 16 T + 130 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.37.q_fa
41$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.41.e_cq
43$D_{4}$ \( 1 + 14 T + 133 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.43.o_fd
47$D_{4}$ \( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.47.m_es
53$D_{4}$ \( 1 - 4 T + 92 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.53.ae_do
59$D_{4}$ \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.59.am_de
61$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.61.c_et
67$D_{4}$ \( 1 + 2 T + 133 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.67.c_fd
71$D_{4}$ \( 1 + 8 T + 140 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.71.i_fk
73$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_fy
79$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \) 2.79.a_ew
83$D_{4}$ \( 1 - 4 T + 98 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.83.ae_du
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.89.i_hm
97$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.97.k_il
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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

−7.45033966146718255745459882505, −7.07142044994554613538679312257, −6.92371300234437675437279032324, −6.64442947262615494371750312881, −6.34338225165280177053042876029, −5.85411183119538066660683511582, −5.27467116677379551236206450093, −5.21898809861469672152245300898, −4.67463536438676540579190492508, −4.32650235703260937147837232197, −3.67426796662008807008777717828, −3.63178786491443892687347116647, −3.28616600778238286471503931330, −3.05236293515755111288527613635, −2.24578310107420008724601955050, −2.07778006675782880969825115239, −1.41575045017063332673814507845, −1.36411332522581255448086182443, 0, 0, 1.36411332522581255448086182443, 1.41575045017063332673814507845, 2.07778006675782880969825115239, 2.24578310107420008724601955050, 3.05236293515755111288527613635, 3.28616600778238286471503931330, 3.63178786491443892687347116647, 3.67426796662008807008777717828, 4.32650235703260937147837232197, 4.67463536438676540579190492508, 5.21898809861469672152245300898, 5.27467116677379551236206450093, 5.85411183119538066660683511582, 6.34338225165280177053042876029, 6.64442947262615494371750312881, 6.92371300234437675437279032324, 7.07142044994554613538679312257, 7.45033966146718255745459882505

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