Properties

Label 4-1470e2-1.1-c1e2-0-9
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 − 2·11-s + 4·13-s − 15-s − 16-s + 4·17-s + 2·22-s − 8·23-s + 24-s − 4·26-s − 27-s + 30-s + 2·31-s − 2·33-s − 4·34-s − 8·37-s + 4·39-s − 40-s − 4·41-s − 4·43-s + 8·46-s − 10·47-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 − 0.603·11-s + 1.10·13-s − 0.258·15-s − 1/4·16-s + 0.970·17-s + 0.426·22-s − 1.66·23-s + 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.182·30-s + 0.359·31-s − 0.348·33-s − 0.685·34-s − 1.31·37-s + 0.640·39-s − 0.158·40-s − 0.624·41-s − 0.609·43-s + 1.17·46-s − 1.45·47-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

\(L(1)\) \(\approx\) \(0.9297065753\)
\(L(\frac12)\) \(\approx\) \(0.9297065753\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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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

−9.859470346948589495423058948290, −9.005603816107176656862476784202, −8.976216198043642749658718548144, −8.479973754203268523874327020104, −8.145483186668815280605286638130, −7.77615168113390262045509967385, −7.64648976897921049246507904540, −6.94431194689249343162199815075, −6.60670607145177418835476222484, −5.88863237856568533960982210817, −5.82316499118906002741641061236, −5.02622196228000690420719415981, −4.75081773305131912254256728444, −3.86508951889876980065559246619, −3.82438806796431671286478404337, −3.14303702469919046431421656439, −2.74482973394057912047619159579, −1.75750546362769892564625976572, −1.55880487216064039821656358376, −0.42196832321326491726024629919, 0.42196832321326491726024629919, 1.55880487216064039821656358376, 1.75750546362769892564625976572, 2.74482973394057912047619159579, 3.14303702469919046431421656439, 3.82438806796431671286478404337, 3.86508951889876980065559246619, 4.75081773305131912254256728444, 5.02622196228000690420719415981, 5.82316499118906002741641061236, 5.88863237856568533960982210817, 6.60670607145177418835476222484, 6.94431194689249343162199815075, 7.64648976897921049246507904540, 7.77615168113390262045509967385, 8.145483186668815280605286638130, 8.479973754203268523874327020104, 8.976216198043642749658718548144, 9.005603816107176656862476784202, 9.859470346948589495423058948290

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