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Properties

Label 4-9600e2-1.1-c1e2-0-9
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·7-s + 3·9-s + 2·13-s − 4·17-s + 2·19-s + 4·21-s + 8·23-s + 4·27-s + 4·29-s − 6·31-s + 4·39-s + 8·41-s − 2·43-s − 4·47-s − 49-s − 8·51-s + 8·53-s + 4·57-s − 4·59-s − 2·61-s + 6·63-s + 18·67-s + 16·69-s + 4·71-s − 8·73-s + 16·79-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s + 9-s + 0.554·13-s − 0.970·17-s + 0.458·19-s + 0.872·21-s + 1.66·23-s + 0.769·27-s + 0.742·29-s − 1.07·31-s + 0.640·39-s + 1.24·41-s − 0.304·43-s − 0.583·47-s − 1/7·49-s − 1.12·51-s + 1.09·53-s + 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.755·63-s + 2.19·67-s + 1.92·69-s + 0.474·71-s − 0.936·73-s + 1.80·79-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: \(0\)
Selberg data: \((4,\ 92160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.743057708\)
\(L(\frac12)\) \(\approx\) \(6.743057708\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.7.ac_f
11$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.11.a_m
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$D_{4}$ \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_bc
19$D_{4}$ \( 1 - 2 T + 29 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_bd
23$D_{4}$ \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_ca
29$D_{4}$ \( 1 - 4 T + 52 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.29.ae_ca
31$D_{4}$ \( 1 + 6 T + 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.31.g_cj
37$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.37.a_bi
41$D_{4}$ \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_dk
43$D_{4}$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.43.c_ad
47$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.47.e_cg
53$D_{4}$ \( 1 - 8 T + 112 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_ei
59$D_{4}$ \( 1 + 4 T + 82 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_de
61$D_{4}$ \( 1 + 2 T + 83 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.61.c_df
67$D_{4}$ \( 1 - 18 T + 205 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.67.as_hx
71$D_{4}$ \( 1 - 4 T + 56 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.71.ae_ce
73$D_{4}$ \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.73.i_es
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$D_{4}$ \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_fa
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.89.i_hm
97$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \) 2.97.be_qd
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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

−7.82784094792066988535550037040, −7.64682142061120779610895770230, −7.10858675724858664091366395008, −7.04587486414889176614526458533, −6.47427367329204083865936727616, −6.41626736670188843121492095385, −5.69661280477772945151162635677, −5.43969665297241996713014407317, −4.91538416383081177052981856887, −4.86114724103786412688754894879, −4.21473206925831982962519561788, −4.09412618070983496645510331741, −3.42484030671897690908607977597, −3.39046966647470865477991545363, −2.62650264807381344932959783288, −2.57565462427304390224918952018, −1.99306736778530496207189526721, −1.52679314605709395667063598461, −1.12085823101960412566889664220, −0.55892316244818492331525412207, 0.55892316244818492331525412207, 1.12085823101960412566889664220, 1.52679314605709395667063598461, 1.99306736778530496207189526721, 2.57565462427304390224918952018, 2.62650264807381344932959783288, 3.39046966647470865477991545363, 3.42484030671897690908607977597, 4.09412618070983496645510331741, 4.21473206925831982962519561788, 4.86114724103786412688754894879, 4.91538416383081177052981856887, 5.43969665297241996713014407317, 5.69661280477772945151162635677, 6.41626736670188843121492095385, 6.47427367329204083865936727616, 7.04587486414889176614526458533, 7.10858675724858664091366395008, 7.64682142061120779610895770230, 7.82784094792066988535550037040

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