Properties

Label 4-3840e2-1.1-c1e2-0-11
Degree $4$
Conductor $14745600$
Sign $1$
Analytic cond. $940.192$
Root an. cond. $5.53737$
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 + 20·37-s + 8·39-s − 4·41-s − 24·43-s − 6·45-s + 10·49-s + 20·53-s + 8·65-s − 16·67-s + 8·71-s + 2·75-s + 16·79-s + 5·81-s − 8·83-s + 12·89-s − 8·107-s − 40·111-s − 12·117-s + 18·121-s + 8·123-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 + 3.28·37-s + 1.28·39-s − 0.624·41-s − 3.65·43-s − 0.894·45-s + 10/7·49-s + 2.74·53-s + 0.992·65-s − 1.95·67-s + 0.949·71-s + 0.230·75-s + 1.80·79-s + 5/9·81-s − 0.878·83-s + 1.27·89-s − 0.773·107-s − 3.79·111-s − 1.10·117-s + 1.63·121-s + 0.721·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(940.192\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6787482142\)
\(L(\frac12)\) \(\approx\) \(0.6787482142\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.740322834163485022205936096313, −8.102577254143068176026485834363, −7.934118290087340528483643131443, −7.38839189272073119099081381015, −7.36755316879797070452781670454, −6.72796400532654923666995763302, −6.57466678806329785547386722874, −6.03727900008917443602226944378, −5.71531446150005638769356676506, −5.26283856781323370648246448306, −4.90353287781904129128496325244, −4.52287740945337835369967547220, −4.27085294422757638879633628040, −3.56232073015672390161557220473, −3.52486949339636624436067475171, −2.49586402847518774850270032372, −2.43855041718298952145646574130, −1.58670216669201233258972177573, −0.915494956748907366418585187790, −0.32908894437133172800253090557, 0.32908894437133172800253090557, 0.915494956748907366418585187790, 1.58670216669201233258972177573, 2.43855041718298952145646574130, 2.49586402847518774850270032372, 3.52486949339636624436067475171, 3.56232073015672390161557220473, 4.27085294422757638879633628040, 4.52287740945337835369967547220, 4.90353287781904129128496325244, 5.26283856781323370648246448306, 5.71531446150005638769356676506, 6.03727900008917443602226944378, 6.57466678806329785547386722874, 6.72796400532654923666995763302, 7.36755316879797070452781670454, 7.38839189272073119099081381015, 7.934118290087340528483643131443, 8.102577254143068176026485834363, 8.740322834163485022205936096313

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