Properties

Label 4-7200e2-1.1-c1e2-0-32
Degree $4$
Conductor $51840000$
Sign $1$
Analytic cond. $3305.36$
Root an. cond. $7.58236$
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  + 10·11-s + 10·19-s + 8·29-s − 20·31-s − 10·41-s + 10·49-s − 20·61-s + 20·79-s − 18·89-s − 4·101-s − 20·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  + 3.01·11-s + 2.29·19-s + 1.48·29-s − 3.59·31-s − 1.56·41-s + 10/7·49-s − 2.56·61-s + 2.25·79-s − 1.90·89-s − 0.398·101-s − 1.91·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(51840000\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(3305.36\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 51840000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.905911058\)
\(L(\frac12)\) \(\approx\) \(3.905911058\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 121 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 165 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−7.893463127453781428065205398208, −7.75935638861024541399912663224, −7.37531108897141827099665843771, −6.97103644968742980111144529460, −6.66817480820214026043513226131, −6.55556994742163320850891697567, −6.04218976959245953152140623554, −5.58733519521942710924459766528, −5.25443699986158129421127384433, −5.11248210463901629885957202798, −4.31334600879724889704186804431, −4.15555517783461570217754808179, −3.72275047826457630248821166216, −3.42169017106700722991706893231, −3.11649486796155157017523672696, −2.55687230888753316463160803997, −1.65196290367657739767103631887, −1.61450444403337656299925740886, −1.18551866908015971922614456941, −0.51092269099572287910079071898, 0.51092269099572287910079071898, 1.18551866908015971922614456941, 1.61450444403337656299925740886, 1.65196290367657739767103631887, 2.55687230888753316463160803997, 3.11649486796155157017523672696, 3.42169017106700722991706893231, 3.72275047826457630248821166216, 4.15555517783461570217754808179, 4.31334600879724889704186804431, 5.11248210463901629885957202798, 5.25443699986158129421127384433, 5.58733519521942710924459766528, 6.04218976959245953152140623554, 6.55556994742163320850891697567, 6.66817480820214026043513226131, 6.97103644968742980111144529460, 7.37531108897141827099665843771, 7.75935638861024541399912663224, 7.893463127453781428065205398208

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