Properties

Label 4-165e2-1.1-c1e2-0-6
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  − 2·3-s − 3·4-s + 2·5-s + 3·9-s − 4·11-s + 6·12-s − 4·15-s + 5·16-s − 6·20-s + 3·25-s − 4·27-s + 8·33-s − 9·36-s − 20·37-s + 12·44-s + 6·45-s + 16·47-s − 10·48-s − 14·49-s − 20·53-s − 8·55-s − 8·59-s + 12·60-s − 3·64-s + 24·67-s − 16·71-s − 6·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 3/2·4-s + 0.894·5-s + 9-s − 1.20·11-s + 1.73·12-s − 1.03·15-s + 5/4·16-s − 1.34·20-s + 3/5·25-s − 0.769·27-s + 1.39·33-s − 3/2·36-s − 3.28·37-s + 1.80·44-s + 0.894·45-s + 2.33·47-s − 1.44·48-s − 2·49-s − 2.74·53-s − 1.07·55-s − 1.04·59-s + 1.54·60-s − 3/8·64-s + 2.93·67-s − 1.89·71-s − 0.692·75-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: $\chi_{27225} (1, \cdot )$
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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
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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

−10.42960000975694463735461863267, −9.724719918800934153926104184181, −9.523451675812284265792380668213, −8.803741276403576480309399510238, −8.307715053946613532072100459003, −7.66488013441745380243230842523, −6.90407371711147677445237484903, −6.32589394889552567488031990655, −5.52379452586844972509172293334, −5.23920392624592057055772361749, −4.82487869342099753310924019144, −4.03102032253133565798645985343, −3.05657152421448946850062568827, −1.66725869558357669408059166263, 0, 1.66725869558357669408059166263, 3.05657152421448946850062568827, 4.03102032253133565798645985343, 4.82487869342099753310924019144, 5.23920392624592057055772361749, 5.52379452586844972509172293334, 6.32589394889552567488031990655, 6.90407371711147677445237484903, 7.66488013441745380243230842523, 8.307715053946613532072100459003, 8.803741276403576480309399510238, 9.523451675812284265792380668213, 9.724719918800934153926104184181, 10.42960000975694463735461863267

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