Properties

Label 4-3840e2-1.1-c1e2-0-47
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 + 4·5-s + 3·9-s + 8·15-s + 11·25-s + 4·27-s − 8·31-s + 16·37-s + 20·41-s − 8·43-s + 12·45-s − 2·49-s − 24·53-s + 8·67-s + 22·75-s − 24·79-s + 5·81-s + 8·83-s − 20·89-s − 16·93-s + 24·107-s + 32·111-s + 6·121-s + 40·123-s + 24·125-s + 127-s − 16·129-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 9-s + 2.06·15-s + 11/5·25-s + 0.769·27-s − 1.43·31-s + 2.63·37-s + 3.12·41-s − 1.21·43-s + 1.78·45-s − 2/7·49-s − 3.29·53-s + 0.977·67-s + 2.54·75-s − 2.70·79-s + 5/9·81-s + 0.878·83-s − 2.11·89-s − 1.65·93-s + 2.32·107-s + 3.03·111-s + 6/11·121-s + 3.60·123-s + 2.14·125-s + 0.0887·127-s − 1.40·129-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

\(L(1)\) \(\approx\) \(7.246460650\)
\(L(\frac12)\) \(\approx\) \(7.246460650\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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

−8.790165115090069543544324857368, −8.276555353635208334176176248068, −8.053489724833257968419918576678, −7.54209363257644188226591206895, −7.30212945367935585218786793098, −6.91750512104070747468753812056, −6.24848184896888642789081534520, −6.04433641931386524583642277522, −5.99125627803011138061045023709, −5.24134973955034089485771777697, −4.96060550476014776285030158685, −4.40843713773104647927832779843, −4.15436479906600123882530211764, −3.54249169822309023329639140240, −2.90110785206999153107689811766, −2.79061836263945101001896086968, −2.32285434847346042853356653305, −1.67853602402807342262235977792, −1.51878371058921089288147181231, −0.70526333358588687860869285987, 0.70526333358588687860869285987, 1.51878371058921089288147181231, 1.67853602402807342262235977792, 2.32285434847346042853356653305, 2.79061836263945101001896086968, 2.90110785206999153107689811766, 3.54249169822309023329639140240, 4.15436479906600123882530211764, 4.40843713773104647927832779843, 4.96060550476014776285030158685, 5.24134973955034089485771777697, 5.99125627803011138061045023709, 6.04433641931386524583642277522, 6.24848184896888642789081534520, 6.91750512104070747468753812056, 7.30212945367935585218786793098, 7.54209363257644188226591206895, 8.053489724833257968419918576678, 8.276555353635208334176176248068, 8.790165115090069543544324857368

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