Properties

Label 4-1320e2-1.1-c1e2-0-28
Degree $4$
Conductor $1742400$
Sign $1$
Analytic cond. $111.096$
Root an. cond. $3.24657$
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 − 4·11-s + 4·15-s + 3·25-s + 4·27-s − 16·31-s − 8·33-s − 4·37-s + 6·45-s + 16·47-s − 14·49-s + 12·53-s − 8·55-s + 24·59-s + 8·67-s + 16·71-s + 6·75-s + 5·81-s + 20·89-s − 32·93-s + 4·97-s − 12·99-s − 8·111-s − 12·113-s + 5·121-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 9-s − 1.20·11-s + 1.03·15-s + 3/5·25-s + 0.769·27-s − 2.87·31-s − 1.39·33-s − 0.657·37-s + 0.894·45-s + 2.33·47-s − 2·49-s + 1.64·53-s − 1.07·55-s + 3.12·59-s + 0.977·67-s + 1.89·71-s + 0.692·75-s + 5/9·81-s + 2.11·89-s − 3.31·93-s + 0.406·97-s − 1.20·99-s − 0.759·111-s − 1.12·113-s + 5/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1742400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(111.096\)
Root analytic conductor: \(3.24657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1742400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.694297949\)
\(L(\frac12)\) \(\approx\) \(3.694297949\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−7.82509924420319171607901005177, −7.47107963029260271307547882484, −6.89294010622960370374517965040, −6.89270810769313104844169224865, −6.01233132932967091954644036311, −5.59585158632682227968586342616, −5.14697097430576642263450693233, −5.00754822907928208673608959123, −3.95303834850677407956821293662, −3.80035613349284838151879027729, −3.23930613694872092422065115869, −2.46356630345198587679515772045, −2.25002432004047770108316173263, −1.77182976748085837883300948740, −0.72522583805620120659533172797, 0.72522583805620120659533172797, 1.77182976748085837883300948740, 2.25002432004047770108316173263, 2.46356630345198587679515772045, 3.23930613694872092422065115869, 3.80035613349284838151879027729, 3.95303834850677407956821293662, 5.00754822907928208673608959123, 5.14697097430576642263450693233, 5.59585158632682227968586342616, 6.01233132932967091954644036311, 6.89270810769313104844169224865, 6.89294010622960370374517965040, 7.47107963029260271307547882484, 7.82509924420319171607901005177

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