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Properties

Label 4-2100e2-1.1-c1e2-0-11
Degree $4$
Conductor $4410000$
Sign $1$
Analytic cond. $281.185$
Root an. cond. $4.09494$
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  − 3·3-s − 7-s + 6·9-s + 9·19-s + 3·21-s − 9·27-s + 18·31-s − 11·37-s + 16·43-s − 6·49-s − 27·57-s − 15·61-s − 6·63-s + 11·67-s + 3·73-s − 17·79-s + 9·81-s − 54·93-s − 27·103-s + 17·109-s + 33·111-s − 11·121-s + 127-s − 48·129-s + 131-s − 9·133-s + 137-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.377·7-s + 2·9-s + 2.06·19-s + 0.654·21-s − 1.73·27-s + 3.23·31-s − 1.80·37-s + 2.43·43-s − 6/7·49-s − 3.57·57-s − 1.92·61-s − 0.755·63-s + 1.34·67-s + 0.351·73-s − 1.91·79-s + 81-s − 5.59·93-s − 2.66·103-s + 1.62·109-s + 3.13·111-s − 121-s + 0.0887·127-s − 4.22·129-s + 0.0873·131-s − 0.780·133-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4410000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(281.185\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4410000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.180462933\)
\(L(\frac12)\) \(\approx\) \(1.180462933\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.11.a_l
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.13.a_ax
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.17.a_ar
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.19.aj_bu
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.23.a_x
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.31.as_fj
37$C_2$ \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.l_dg
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.53.a_cb
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.p_fg
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.67.al_cc
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.73.ad_cy
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.r_ic
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.89.a_adl
97$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.a_gn
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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

−9.581950899791057790654591589964, −8.918578397705236237280308743391, −8.603076792890712723112962449592, −7.83507468217309468198889740798, −7.82882955143568715616064285461, −7.21071891099072531343893674178, −6.76018892118948254980797444285, −6.65743859286813425118761954100, −6.01646777509454074820062745265, −5.77998359134134617036402373952, −5.41934086171437370720867397585, −4.87224034530790931437938025971, −4.65470576671331480995977807962, −4.15162543046267556401927410788, −3.57999959231947949951621920204, −2.95162090774115580860607682871, −2.64013528603393592114690897856, −1.58183247522925483768296780285, −1.10309200269745362228949041035, −0.52215734284593936210069224337, 0.52215734284593936210069224337, 1.10309200269745362228949041035, 1.58183247522925483768296780285, 2.64013528603393592114690897856, 2.95162090774115580860607682871, 3.57999959231947949951621920204, 4.15162543046267556401927410788, 4.65470576671331480995977807962, 4.87224034530790931437938025971, 5.41934086171437370720867397585, 5.77998359134134617036402373952, 6.01646777509454074820062745265, 6.65743859286813425118761954100, 6.76018892118948254980797444285, 7.21071891099072531343893674178, 7.82882955143568715616064285461, 7.83507468217309468198889740798, 8.603076792890712723112962449592, 8.918578397705236237280308743391, 9.581950899791057790654591589964

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