Properties

Label 4-165e2-1.1-c1e2-0-13
Degree $4$
Conductor $27225$
Sign $-1$
Analytic cond. $1.73588$
Root an. cond. $1.14783$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4-s − 3·5-s − 2·9-s − 12-s − 3·15-s − 3·16-s + 3·20-s − 12·23-s + 4·25-s − 5·27-s − 5·31-s + 2·36-s + 10·37-s + 6·45-s + 6·47-s − 3·48-s − 4·49-s − 3·53-s + 3·60-s + 7·64-s − 5·67-s − 12·69-s − 15·71-s + 4·75-s + 9·80-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 1/2·4-s − 1.34·5-s − 2/3·9-s − 0.288·12-s − 0.774·15-s − 3/4·16-s + 0.670·20-s − 2.50·23-s + 4/5·25-s − 0.962·27-s − 0.898·31-s + 1/3·36-s + 1.64·37-s + 0.894·45-s + 0.875·47-s − 0.433·48-s − 4/7·49-s − 0.412·53-s + 0.387·60-s + 7/8·64-s − 0.610·67-s − 1.44·69-s − 1.78·71-s + 0.461·75-s + 1.00·80-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27225\)    =    \(3^{2} \cdot 5^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1.73588\)
Root analytic conductor: \(1.14783\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 27225,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_2$ \( 1 - T + p T^{2} \)
5$C_2$ \( 1 + 3 T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
31$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 101 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
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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

−10.30982857159085311297526685253, −9.697541331920453929411464213255, −9.137689966494638278419622499989, −8.733996668882291463947112532613, −8.134409316591164439222137062358, −7.73619965668430178372809436080, −7.41167413887269899059610551273, −6.37144203420924223332184025187, −5.92076060987708211584074184496, −5.04550762825246398726288248415, −4.12981830041920485037693648993, −4.01309760116970206841172254780, −3.07470313069148291546187167441, −2.14164777723408462592642893404, 0, 2.14164777723408462592642893404, 3.07470313069148291546187167441, 4.01309760116970206841172254780, 4.12981830041920485037693648993, 5.04550762825246398726288248415, 5.92076060987708211584074184496, 6.37144203420924223332184025187, 7.41167413887269899059610551273, 7.73619965668430178372809436080, 8.134409316591164439222137062358, 8.733996668882291463947112532613, 9.137689966494638278419622499989, 9.697541331920453929411464213255, 10.30982857159085311297526685253

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