Properties

Label 4-1470e2-1.1-c1e2-0-8
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 − 3·11-s − 10·13-s − 15-s − 16-s + 5·19-s − 3·22-s + 9·23-s − 24-s − 10·26-s − 27-s − 30-s − 10·31-s − 3·33-s + 37-s + 5·38-s − 10·39-s + 40-s − 18·41-s + 16·43-s + 9·46-s + 3·47-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 − 0.904·11-s − 2.77·13-s − 0.258·15-s − 1/4·16-s + 1.14·19-s − 0.639·22-s + 1.87·23-s − 0.204·24-s − 1.96·26-s − 0.192·27-s − 0.182·30-s − 1.79·31-s − 0.522·33-s + 0.164·37-s + 0.811·38-s − 1.60·39-s + 0.158·40-s − 2.81·41-s + 2.43·43-s + 1.32·46-s + 0.437·47-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2160900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.493675864\)
\(L(\frac12)\) \(\approx\) \(1.493675864\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
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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

−9.583349445371523017946238017070, −9.338952788673778718278910018398, −8.900167007576851669182415281404, −8.639786271306232522052338089798, −7.86690420206183059297713685927, −7.62201900451264564953591475851, −7.40222331275373960640187663321, −6.83306942599835031352132826012, −6.81029359105270310955149368595, −5.63271476678346637420463689687, −5.42140972191239109701616912053, −5.08915798257321739408074917488, −4.88629136510676719090207660902, −4.04767663942300464093416786516, −3.87074536342218035859977059138, −2.91854545247552529900806380580, −2.87091377713771768076655416324, −2.43240904203746776925018671566, −1.57234960949782816263250875200, −0.40367503742902975301103920312, 0.40367503742902975301103920312, 1.57234960949782816263250875200, 2.43240904203746776925018671566, 2.87091377713771768076655416324, 2.91854545247552529900806380580, 3.87074536342218035859977059138, 4.04767663942300464093416786516, 4.88629136510676719090207660902, 5.08915798257321739408074917488, 5.42140972191239109701616912053, 5.63271476678346637420463689687, 6.81029359105270310955149368595, 6.83306942599835031352132826012, 7.40222331275373960640187663321, 7.62201900451264564953591475851, 7.86690420206183059297713685927, 8.639786271306232522052338089798, 8.900167007576851669182415281404, 9.338952788673778718278910018398, 9.583349445371523017946238017070

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