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Properties

Label 4-960e2-1.1-c1e2-0-31
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 + 3·9-s + 4·13-s − 4·15-s − 25-s − 4·27-s + 16·31-s + 20·37-s − 8·39-s − 12·41-s + 8·43-s + 6·45-s + 10·49-s + 20·53-s + 8·65-s − 24·67-s + 2·75-s + 32·79-s + 5·81-s + 8·83-s + 20·89-s − 32·93-s − 40·107-s − 40·111-s + 12·117-s + 18·121-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 9-s + 1.10·13-s − 1.03·15-s − 1/5·25-s − 0.769·27-s + 2.87·31-s + 3.28·37-s − 1.28·39-s − 1.87·41-s + 1.21·43-s + 0.894·45-s + 10/7·49-s + 2.74·53-s + 0.992·65-s − 2.93·67-s + 0.230·75-s + 3.60·79-s + 5/9·81-s + 0.878·83-s + 2.11·89-s − 3.31·93-s − 3.86·107-s − 3.79·111-s + 1.10·117-s + 1.63·121-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

\(L(1)\) \(\approx\) \(2.053070271\)
\(L(\frac12)\) \(\approx\) \(2.053070271\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.19.a_abi
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.a_abq
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.59.a_ade
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.67.y_ks
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
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.ai_ha
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.97.a_ck
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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

−10.31761260943870157738625025801, −9.944150576924295454554433021481, −9.316612297881715635610248252856, −9.311777268275246627115943639247, −8.481274652820164988815424914377, −8.234025792273744250414436771549, −7.64814639973234439515610557836, −7.24960040314863171584075486849, −6.53311196146945867908972306599, −6.32643152367091625855879949501, −5.85859111757882876564039662290, −5.81459289361072466628722626456, −4.89993980612448765572676202232, −4.72559186444712509004153418741, −4.02680326529006076127917706063, −3.64267859417867646092280427371, −2.54811782788153196829914844145, −2.38800983033393017202192271592, −1.18778929982582805162275796947, −0.884294251367325343545835127185, 0.884294251367325343545835127185, 1.18778929982582805162275796947, 2.38800983033393017202192271592, 2.54811782788153196829914844145, 3.64267859417867646092280427371, 4.02680326529006076127917706063, 4.72559186444712509004153418741, 4.89993980612448765572676202232, 5.81459289361072466628722626456, 5.85859111757882876564039662290, 6.32643152367091625855879949501, 6.53311196146945867908972306599, 7.24960040314863171584075486849, 7.64814639973234439515610557836, 8.234025792273744250414436771549, 8.481274652820164988815424914377, 9.311777268275246627115943639247, 9.316612297881715635610248252856, 9.944150576924295454554433021481, 10.31761260943870157738625025801

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