Properties

Label 4-180e2-1.1-c1e2-0-1
Degree $4$
Conductor $32400$
Sign $1$
Analytic cond. $2.06585$
Root an. cond. $1.19887$
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  − 2·5-s + 8·11-s − 25-s − 12·29-s + 8·31-s + 20·41-s − 2·49-s − 16·55-s + 8·59-s + 4·61-s + 24·79-s − 20·89-s + 4·101-s + 4·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s − 16·155-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.894·5-s + 2.41·11-s − 1/5·25-s − 2.22·29-s + 1.43·31-s + 3.12·41-s − 2/7·49-s − 2.15·55-s + 1.04·59-s + 0.512·61-s + 2.70·79-s − 2.11·89-s + 0.398·101-s + 0.383·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s − 1.28·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(32400\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2.06585\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{180} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 32400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.207743441\)
\(L(\frac12)\) \(\approx\) \(1.207743441\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−12.60025063249490897346720021931, −12.54147526348877983719269710054, −11.69464799482469199651062459216, −11.52645430686467146683683686669, −11.21489140015531959037072666364, −10.55163908123743082372848797502, −9.658139086457238696235799681109, −9.458454479382057127776008723656, −8.949372517253268092969552758100, −8.382882220249915634657648452507, −7.64201655135141947135104127120, −7.38886104694848865058450009257, −6.54357012282433405503891675422, −6.23353108455923494477092678570, −5.47840540371397839884954858337, −4.51725979962021635841320920932, −3.81805801479417206425639861289, −3.78070346781804821885924275094, −2.41444319834448939407476003740, −1.15980437545413181859700964876, 1.15980437545413181859700964876, 2.41444319834448939407476003740, 3.78070346781804821885924275094, 3.81805801479417206425639861289, 4.51725979962021635841320920932, 5.47840540371397839884954858337, 6.23353108455923494477092678570, 6.54357012282433405503891675422, 7.38886104694848865058450009257, 7.64201655135141947135104127120, 8.382882220249915634657648452507, 8.949372517253268092969552758100, 9.458454479382057127776008723656, 9.658139086457238696235799681109, 10.55163908123743082372848797502, 11.21489140015531959037072666364, 11.52645430686467146683683686669, 11.69464799482469199651062459216, 12.54147526348877983719269710054, 12.60025063249490897346720021931

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