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Properties

Label 4-960e2-1.1-c1e2-0-5
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 9-s − 12·11-s − 8·19-s + 11·25-s + 16·31-s − 12·41-s − 4·45-s + 10·49-s − 48·55-s + 12·59-s + 12·61-s + 8·71-s − 16·79-s + 81-s − 28·89-s − 32·95-s + 12·99-s − 16·101-s + 4·109-s + 86·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s − 3.61·11-s − 1.83·19-s + 11/5·25-s + 2.87·31-s − 1.87·41-s − 0.596·45-s + 10/7·49-s − 6.47·55-s + 1.56·59-s + 1.53·61-s + 0.949·71-s − 1.80·79-s + 1/9·81-s − 2.96·89-s − 3.28·95-s + 1.20·99-s − 1.59·101-s + 0.383·109-s + 7.81·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-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

\(L(1)\) \(\approx\) \(1.648629722\)
\(L(\frac12)\) \(\approx\) \(1.648629722\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.47.a_ada
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.61.am_gc
67$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.67.a_afe
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.73.a_ac
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−10.16301143950663709372774734815, −10.10843074733092975834423502054, −9.677742935051225656092075022861, −8.824488093944440135428863578877, −8.455995937978732418884194336711, −8.277596149553374920068614923576, −7.968926470881624564798648519485, −7.18008565536287803926850182462, −6.72118002178376890734929170176, −6.44912600717094385911057205493, −5.68519514885472871063497048270, −5.45498881352701941400877597056, −5.28413821514658668311735752097, −4.62606190514689230156777348903, −4.18954719814143513552709798199, −2.91049022848173918235475093563, −2.80070665029808225874580405786, −2.35021042146029618209966128923, −1.85377158794925610007949732812, −0.55855792703317023892837593089, 0.55855792703317023892837593089, 1.85377158794925610007949732812, 2.35021042146029618209966128923, 2.80070665029808225874580405786, 2.91049022848173918235475093563, 4.18954719814143513552709798199, 4.62606190514689230156777348903, 5.28413821514658668311735752097, 5.45498881352701941400877597056, 5.68519514885472871063497048270, 6.44912600717094385911057205493, 6.72118002178376890734929170176, 7.18008565536287803926850182462, 7.968926470881624564798648519485, 8.277596149553374920068614923576, 8.455995937978732418884194336711, 8.824488093944440135428863578877, 9.677742935051225656092075022861, 10.10843074733092975834423502054, 10.16301143950663709372774734815

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