Properties

Label 4-165e2-1.1-c1e2-0-2
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 $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s − 2·5-s + 4·7-s + 3·9-s − 2·11-s − 2·12-s + 4·13-s − 4·15-s − 3·16-s + 4·19-s + 2·20-s + 8·21-s + 3·25-s + 4·27-s − 4·28-s − 8·31-s − 4·33-s − 8·35-s − 3·36-s + 4·37-s + 8·39-s + 4·43-s + 2·44-s − 6·45-s − 6·48-s − 2·49-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 0.894·5-s + 1.51·7-s + 9-s − 0.603·11-s − 0.577·12-s + 1.10·13-s − 1.03·15-s − 3/4·16-s + 0.917·19-s + 0.447·20-s + 1.74·21-s + 3/5·25-s + 0.769·27-s − 0.755·28-s − 1.43·31-s − 0.696·33-s − 1.35·35-s − 1/2·36-s + 0.657·37-s + 1.28·39-s + 0.609·43-s + 0.301·44-s − 0.894·45-s − 0.866·48-s − 2/7·49-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: induced by $\chi_{165} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 27225,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.616317410\)
\(L(\frac12)\) \(\approx\) \(1.616317410\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−12.93773955410003035298850172344, −12.89752814607709570945352996584, −12.12442844884684652899375597193, −11.30334399265785332516407629334, −11.19000972668998539849998229463, −10.85392291262864832309776636525, −9.835971336260779695431693116780, −9.499164343979236981260608269032, −8.716304766367879631910908910682, −8.538147174097694339979543049872, −7.938681887368569925608943291661, −7.64462765613727710558979979432, −7.08359108028221100742001111731, −6.18401801978408377537006310704, −5.13149880959224774315664303114, −4.84301865879241464751162615703, −3.99060451599707408140172652399, −3.56028157284573386243169694421, −2.54117248937871306304050278597, −1.45710137320084150621502204244, 1.45710137320084150621502204244, 2.54117248937871306304050278597, 3.56028157284573386243169694421, 3.99060451599707408140172652399, 4.84301865879241464751162615703, 5.13149880959224774315664303114, 6.18401801978408377537006310704, 7.08359108028221100742001111731, 7.64462765613727710558979979432, 7.938681887368569925608943291661, 8.538147174097694339979543049872, 8.716304766367879631910908910682, 9.499164343979236981260608269032, 9.835971336260779695431693116780, 10.85392291262864832309776636525, 11.19000972668998539849998229463, 11.30334399265785332516407629334, 12.12442844884684652899375597193, 12.89752814607709570945352996584, 12.93773955410003035298850172344

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