Properties

Label 4-3870e2-1.1-c1e2-0-14
Degree $4$
Conductor $14976900$
Sign $1$
Analytic cond. $954.940$
Root an. cond. $5.55896$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 2·5-s + 2·7-s − 4·8-s + 4·10-s − 4·11-s + 2·13-s − 4·14-s + 5·16-s − 2·19-s − 6·20-s + 8·22-s − 4·23-s + 3·25-s − 4·26-s + 6·28-s + 6·29-s + 2·31-s − 6·32-s − 4·35-s − 4·37-s + 4·38-s + 8·40-s − 2·41-s + 2·43-s − 12·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s − 1.41·8-s + 1.26·10-s − 1.20·11-s + 0.554·13-s − 1.06·14-s + 5/4·16-s − 0.458·19-s − 1.34·20-s + 1.70·22-s − 0.834·23-s + 3/5·25-s − 0.784·26-s + 1.13·28-s + 1.11·29-s + 0.359·31-s − 1.06·32-s − 0.676·35-s − 0.657·37-s + 0.648·38-s + 1.26·40-s − 0.312·41-s + 0.304·43-s − 1.80·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14976900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14976900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(954.940\)
Root analytic conductor: \(5.55896\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 14976900,\ (\ :1/2, 1/2),\ 1)\)

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)$
bad2$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
43$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$D_{4}$ \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 49 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 61 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 2 T + 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 14 T + 121 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 125 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 10 T + 153 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 14 T + 205 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 132 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T - 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−8.241669739870149283836674822940, −8.143741381868653651411108854926, −7.67117807827751063418815108658, −7.45585138471127397972734304308, −6.90126852023982865703373361414, −6.71266218213201857499482988792, −6.02187668020161944507879365484, −6.00687014121768021379130874282, −5.17723864549500521165003614427, −5.06499619059360628893298725523, −4.34871104342274581425942315860, −4.22122069157302799024294649704, −3.33780698377096100173597973641, −3.25459486916923672932415915070, −2.39693345549122766359110420989, −2.35665957952903390817281418107, −1.30931584673592249335107008659, −1.29815046478113059533226265104, 0, 0, 1.29815046478113059533226265104, 1.30931584673592249335107008659, 2.35665957952903390817281418107, 2.39693345549122766359110420989, 3.25459486916923672932415915070, 3.33780698377096100173597973641, 4.22122069157302799024294649704, 4.34871104342274581425942315860, 5.06499619059360628893298725523, 5.17723864549500521165003614427, 6.00687014121768021379130874282, 6.02187668020161944507879365484, 6.71266218213201857499482988792, 6.90126852023982865703373361414, 7.45585138471127397972734304308, 7.67117807827751063418815108658, 8.143741381868653651411108854926, 8.241669739870149283836674822940

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