Properties

Label 4-6240e2-1.1-c1e2-0-8
Degree $4$
Conductor $38937600$
Sign $1$
Analytic cond. $2482.69$
Root an. cond. $7.05879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 3·9-s + 2·13-s + 4·15-s + 4·17-s − 8·19-s + 8·23-s + 3·25-s − 4·27-s − 4·29-s − 8·31-s − 4·37-s − 4·39-s − 4·41-s − 8·43-s − 6·45-s − 8·47-s − 6·49-s − 8·51-s − 4·53-s + 16·57-s + 12·61-s − 4·65-s − 16·69-s + 16·71-s + 20·73-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 9-s + 0.554·13-s + 1.03·15-s + 0.970·17-s − 1.83·19-s + 1.66·23-s + 3/5·25-s − 0.769·27-s − 0.742·29-s − 1.43·31-s − 0.657·37-s − 0.640·39-s − 0.624·41-s − 1.21·43-s − 0.894·45-s − 1.16·47-s − 6/7·49-s − 1.12·51-s − 0.549·53-s + 2.11·57-s + 1.53·61-s − 0.496·65-s − 1.92·69-s + 1.89·71-s + 2.34·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(38937600\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(2482.69\)
Root analytic conductor: \(7.05879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 38937600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
73$C_4$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 174 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−7.72110541379366259502807123142, −7.71878625959832598377336505726, −6.84775307139191924109424521523, −6.82024108839028276294187812984, −6.47182732253031108130050905720, −6.38749609240752239953024993449, −5.51537177124581582668969735735, −5.33793574754763320591074739203, −4.97269078446986327272435223134, −4.93845505461745544324141015095, −4.05693533066938727376903134210, −3.92170496559169759065824575125, −3.46121125390384002069196025271, −3.31207177792891291363343491148, −2.43479526985920936623784068250, −2.03406364065911879725986757058, −1.35705521936650401298490173657, −1.04164016830613767829906517880, 0, 0, 1.04164016830613767829906517880, 1.35705521936650401298490173657, 2.03406364065911879725986757058, 2.43479526985920936623784068250, 3.31207177792891291363343491148, 3.46121125390384002069196025271, 3.92170496559169759065824575125, 4.05693533066938727376903134210, 4.93845505461745544324141015095, 4.97269078446986327272435223134, 5.33793574754763320591074739203, 5.51537177124581582668969735735, 6.38749609240752239953024993449, 6.47182732253031108130050905720, 6.82024108839028276294187812984, 6.84775307139191924109424521523, 7.71878625959832598377336505726, 7.72110541379366259502807123142

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