Properties

Label 4-1620e2-1.1-c1e2-0-6
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $167.334$
Root an. cond. $3.59663$
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  + 5-s − 2·7-s − 2·13-s + 6·17-s + 10·19-s + 3·23-s − 6·29-s − 5·31-s − 2·35-s + 4·37-s + 12·41-s − 8·43-s − 12·47-s + 7·49-s + 6·53-s + 6·59-s + 7·61-s − 2·65-s − 2·67-s − 24·71-s − 32·73-s + 79-s − 15·83-s + 6·85-s + 24·89-s + 4·91-s + 10·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.755·7-s − 0.554·13-s + 1.45·17-s + 2.29·19-s + 0.625·23-s − 1.11·29-s − 0.898·31-s − 0.338·35-s + 0.657·37-s + 1.87·41-s − 1.21·43-s − 1.75·47-s + 49-s + 0.824·53-s + 0.781·59-s + 0.896·61-s − 0.248·65-s − 0.244·67-s − 2.84·71-s − 3.74·73-s + 0.112·79-s − 1.64·83-s + 0.650·85-s + 2.54·89-s + 0.419·91-s + 1.02·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(167.334\)
Root analytic conductor: \(3.59663\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.323717615\)
\(L(\frac12)\) \(\approx\) \(2.323717615\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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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

−9.627096946625867461752248566673, −9.229878683424296962663505641322, −8.942810603845708943826866275967, −8.540608620559694123648541416883, −7.79115672846270028165862531331, −7.42353419162335916711038543214, −7.34593049514981749829029524624, −7.02402798258473775936452474587, −6.15296500148472383129224226925, −5.86131873523810069205181729905, −5.64784628382244551014933536032, −5.15833923064090687449860109703, −4.70216987705528096228663152106, −4.11685894741917049189733146477, −3.31939411859317928661593182603, −3.28028393323370994930025626178, −2.78976220853864652860273553189, −1.96884560660324572121077180159, −1.34170110184150463463131333010, −0.63043147536621173449072760429, 0.63043147536621173449072760429, 1.34170110184150463463131333010, 1.96884560660324572121077180159, 2.78976220853864652860273553189, 3.28028393323370994930025626178, 3.31939411859317928661593182603, 4.11685894741917049189733146477, 4.70216987705528096228663152106, 5.15833923064090687449860109703, 5.64784628382244551014933536032, 5.86131873523810069205181729905, 6.15296500148472383129224226925, 7.02402798258473775936452474587, 7.34593049514981749829029524624, 7.42353419162335916711038543214, 7.79115672846270028165862531331, 8.540608620559694123648541416883, 8.942810603845708943826866275967, 9.229878683424296962663505641322, 9.627096946625867461752248566673

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