Properties

Label 4-165e2-1.1-c1e2-0-9
Degree $4$
Conductor $27225$
Sign $-1$
Analytic cond. $1.73588$
Root an. cond. $1.14783$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 2·5-s + 9-s + 2·11-s + 5·16-s + 6·20-s − 25-s − 12·29-s − 16·31-s − 3·36-s − 4·41-s − 6·44-s − 2·45-s + 2·49-s − 4·55-s − 8·59-s + 12·61-s − 3·64-s − 8·79-s − 10·80-s + 81-s − 12·89-s + 2·99-s + 3·100-s + 4·101-s − 4·109-s + 36·116-s + ⋯
L(s)  = 1  − 3/2·4-s − 0.894·5-s + 1/3·9-s + 0.603·11-s + 5/4·16-s + 1.34·20-s − 1/5·25-s − 2.22·29-s − 2.87·31-s − 1/2·36-s − 0.624·41-s − 0.904·44-s − 0.298·45-s + 2/7·49-s − 0.539·55-s − 1.04·59-s + 1.53·61-s − 3/8·64-s − 0.900·79-s − 1.11·80-s + 1/9·81-s − 1.27·89-s + 0.201·99-s + 3/10·100-s + 0.398·101-s − 0.383·109-s + 3.34·116-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: no
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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−10.17772545901372850796013638635, −9.705159010656125332480920056237, −9.083856298436522884927706897826, −8.975261543279013968212899152873, −8.281293500899591134516470178784, −7.46995816184461358823952634116, −7.43398897048000830563917400761, −6.48836008647392510605014123287, −5.45807303096955976252785458823, −5.31622482563829752699303340190, −4.20407903283329927801590362766, −3.95949329332453626615268855815, −3.41592277941264934062420800268, −1.76559017524548890655133359279, 0, 1.76559017524548890655133359279, 3.41592277941264934062420800268, 3.95949329332453626615268855815, 4.20407903283329927801590362766, 5.31622482563829752699303340190, 5.45807303096955976252785458823, 6.48836008647392510605014123287, 7.43398897048000830563917400761, 7.46995816184461358823952634116, 8.281293500899591134516470178784, 8.975261543279013968212899152873, 9.083856298436522884927706897826, 9.705159010656125332480920056237, 10.17772545901372850796013638635

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