Properties

Label 4-1470e2-1.1-c1e2-0-6
Degree $4$
Conductor $2160900$
Sign $1$
Analytic cond. $137.780$
Root an. cond. $3.42607$
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-s − 3-s − 5-s − 6-s − 8-s − 10-s − 6·11-s + 12·13-s + 15-s − 16-s + 4·19-s − 6·22-s + 24-s + 12·26-s + 27-s − 16·29-s + 30-s − 2·31-s + 6·33-s − 4·37-s + 4·38-s − 12·39-s + 40-s − 20·41-s − 12·43-s + 2·47-s + 48-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 0.447·5-s − 0.408·6-s − 0.353·8-s − 0.316·10-s − 1.80·11-s + 3.32·13-s + 0.258·15-s − 1/4·16-s + 0.917·19-s − 1.27·22-s + 0.204·24-s + 2.35·26-s + 0.192·27-s − 2.97·29-s + 0.182·30-s − 0.359·31-s + 1.04·33-s − 0.657·37-s + 0.648·38-s − 1.92·39-s + 0.158·40-s − 3.12·41-s − 1.82·43-s + 0.291·47-s + 0.144·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2160900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(137.780\)
Root analytic conductor: \(3.42607\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2160900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.043393454\)
\(L(\frac12)\) \(\approx\) \(1.043393454\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good11$C_2^2$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
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

−9.850772558168846196110030715334, −9.075030076268185221531080378088, −8.955466640314035603967422821677, −8.392065745180742976954026678755, −8.206743980650072918043135650299, −7.52596873001703172416476315627, −7.49262597290169564919552252457, −6.61474828444259376711934654195, −6.40454443889734962686846198108, −5.80885774896028068653041968795, −5.63199499408319090999921245342, −5.05015336255964301849953312111, −4.98180200108526678902327097794, −4.11186049711831909206671923886, −3.55890901417255229017515382962, −3.37406758071836814924915307209, −3.11034124194733213430754189262, −1.70920117002487773064211473052, −1.70288645393469267938878482188, −0.37378416786152686106688979303, 0.37378416786152686106688979303, 1.70288645393469267938878482188, 1.70920117002487773064211473052, 3.11034124194733213430754189262, 3.37406758071836814924915307209, 3.55890901417255229017515382962, 4.11186049711831909206671923886, 4.98180200108526678902327097794, 5.05015336255964301849953312111, 5.63199499408319090999921245342, 5.80885774896028068653041968795, 6.40454443889734962686846198108, 6.61474828444259376711934654195, 7.49262597290169564919552252457, 7.52596873001703172416476315627, 8.206743980650072918043135650299, 8.392065745180742976954026678755, 8.955466640314035603967422821677, 9.075030076268185221531080378088, 9.850772558168846196110030715334

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