Properties

Label 4-230e2-1.1-c1e2-0-1
Degree $4$
Conductor $52900$
Sign $1$
Analytic cond. $3.37294$
Root an. cond. $1.35519$
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  + 4-s + 4·5-s − 6·9-s + 4·11-s + 16-s − 4·19-s + 4·20-s + 11·25-s + 4·29-s − 6·36-s + 12·41-s + 4·44-s − 24·45-s + 2·49-s + 16·55-s + 24·59-s − 16·61-s + 64-s − 4·76-s − 24·79-s + 4·80-s + 27·81-s − 12·89-s − 16·95-s − 24·99-s + 11·100-s − 20·101-s + ⋯
L(s)  = 1  + 1/2·4-s + 1.78·5-s − 2·9-s + 1.20·11-s + 1/4·16-s − 0.917·19-s + 0.894·20-s + 11/5·25-s + 0.742·29-s − 36-s + 1.87·41-s + 0.603·44-s − 3.57·45-s + 2/7·49-s + 2.15·55-s + 3.12·59-s − 2.04·61-s + 1/8·64-s − 0.458·76-s − 2.70·79-s + 0.447·80-s + 3·81-s − 1.27·89-s − 1.64·95-s − 2.41·99-s + 1.09·100-s − 1.99·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(52900\)    =    \(2^{2} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(3.37294\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{52900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 52900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.953403245\)
\(L(\frac12)\) \(\approx\) \(1.953403245\)
\(L(\frac{3}{2})\) 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$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
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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

−9.928280836928168979628777920327, −9.594811427439617650906242640583, −8.971935452602447588346290189052, −8.651590213691851109739649270908, −8.276543750439528714060318536349, −7.29468472679713756407763618174, −6.69441913641732873591204403077, −6.21475081018625957585151869260, −5.84432417959727908710252168629, −5.50098265982593501879266203197, −4.63556271802396925268184937965, −3.74894127876684642815312676386, −2.63232737474552247848840602025, −2.53491101319297313540699620659, −1.35564478109443112808861515260, 1.35564478109443112808861515260, 2.53491101319297313540699620659, 2.63232737474552247848840602025, 3.74894127876684642815312676386, 4.63556271802396925268184937965, 5.50098265982593501879266203197, 5.84432417959727908710252168629, 6.21475081018625957585151869260, 6.69441913641732873591204403077, 7.29468472679713756407763618174, 8.276543750439528714060318536349, 8.651590213691851109739649270908, 8.971935452602447588346290189052, 9.594811427439617650906242640583, 9.928280836928168979628777920327

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