Properties

Label 4-2340e2-1.1-c1e2-0-2
Degree $4$
Conductor $5475600$
Sign $1$
Analytic cond. $349.129$
Root an. cond. $4.32261$
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  − 4·5-s − 8·7-s + 4·11-s − 6·13-s − 6·17-s + 4·19-s − 4·23-s + 11·25-s + 32·35-s − 8·37-s + 14·41-s − 16·43-s + 8·47-s + 34·49-s + 14·53-s − 16·55-s + 20·59-s + 24·65-s + 16·71-s − 32·77-s + 32·83-s + 24·85-s − 14·89-s + 48·91-s − 16·95-s + 12·103-s + 24·107-s + ⋯
L(s)  = 1  − 1.78·5-s − 3.02·7-s + 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s + 11/5·25-s + 5.40·35-s − 1.31·37-s + 2.18·41-s − 2.43·43-s + 1.16·47-s + 34/7·49-s + 1.92·53-s − 2.15·55-s + 2.60·59-s + 2.97·65-s + 1.89·71-s − 3.64·77-s + 3.51·83-s + 2.60·85-s − 1.48·89-s + 5.03·91-s − 1.64·95-s + 1.18·103-s + 2.32·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5475600\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(349.129\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5475600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6528257399\)
\(L(\frac12)\) \(\approx\) \(0.6528257399\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2^2$ \( 1 + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−9.105396186993257068585171424032, −8.926032643371771807066007268240, −8.495085274284798476093459736567, −8.050448779032864152808405759614, −7.33830184942043699395231898851, −7.20613114473201374929095604989, −6.85345074360539202771821235038, −6.78242932188826588277702428467, −6.06330833002352448655454551968, −5.95692402237381641419598458965, −4.98911975947275714817582857684, −4.88254915731970692211552497139, −4.01182135725353023197460861929, −3.90195398688430907611887589779, −3.47663098656798031285403818845, −3.24821677265466540154949836720, −2.35103085513481078882342672188, −2.33494398314945500993472276994, −0.68083492071063780409107803020, −0.44908064115229555464803894350, 0.44908064115229555464803894350, 0.68083492071063780409107803020, 2.33494398314945500993472276994, 2.35103085513481078882342672188, 3.24821677265466540154949836720, 3.47663098656798031285403818845, 3.90195398688430907611887589779, 4.01182135725353023197460861929, 4.88254915731970692211552497139, 4.98911975947275714817582857684, 5.95692402237381641419598458965, 6.06330833002352448655454551968, 6.78242932188826588277702428467, 6.85345074360539202771821235038, 7.20613114473201374929095604989, 7.33830184942043699395231898851, 8.050448779032864152808405759614, 8.495085274284798476093459736567, 8.926032643371771807066007268240, 9.105396186993257068585171424032

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