Properties

Label 4-1620e2-1.1-c1e2-0-10
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 6·11-s + 13-s − 2·19-s + 6·23-s + 6·29-s − 8·31-s + 35-s − 14·37-s − 6·41-s + 4·43-s + 12·47-s + 7·49-s + 12·53-s + 6·55-s − 11·61-s + 65-s + 7·67-s + 12·71-s + 22·73-s + 6·77-s + 79-s + 6·83-s + 24·89-s + 91-s − 2·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.80·11-s + 0.277·13-s − 0.458·19-s + 1.25·23-s + 1.11·29-s − 1.43·31-s + 0.169·35-s − 2.30·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 49-s + 1.64·53-s + 0.809·55-s − 1.40·61-s + 0.124·65-s + 0.855·67-s + 1.42·71-s + 2.57·73-s + 0.683·77-s + 0.112·79-s + 0.658·83-s + 2.54·89-s + 0.104·91-s − 0.205·95-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

\(L(1)\) \(\approx\) \(3.478193664\)
\(L(\frac12)\) \(\approx\) \(3.478193664\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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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

−9.275862770202948730539458817071, −9.117103165356682067205774154988, −8.896083244660351379071625984535, −8.707507576078659200805731718439, −7.948433025806952808223705155257, −7.64949770793006575000847689344, −6.93452850895556673355372459868, −6.88927875059497253503524900796, −6.43878483426655247084553947299, −6.03255574087117669283223451061, −5.39609020589636824390682936124, −5.13815789471283498099003027906, −4.68973545747129347315240438732, −4.01050746590354684390878290993, −3.56683067622703208294517785351, −3.46317722579935583968584931004, −2.28209736958848734266206952127, −2.16232661081501294189568825430, −1.29058377336987598391147609536, −0.817107985060873369225355709621, 0.817107985060873369225355709621, 1.29058377336987598391147609536, 2.16232661081501294189568825430, 2.28209736958848734266206952127, 3.46317722579935583968584931004, 3.56683067622703208294517785351, 4.01050746590354684390878290993, 4.68973545747129347315240438732, 5.13815789471283498099003027906, 5.39609020589636824390682936124, 6.03255574087117669283223451061, 6.43878483426655247084553947299, 6.88927875059497253503524900796, 6.93452850895556673355372459868, 7.64949770793006575000847689344, 7.948433025806952808223705155257, 8.707507576078659200805731718439, 8.896083244660351379071625984535, 9.117103165356682067205774154988, 9.275862770202948730539458817071

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