Properties

Label 4-1620e2-1.1-c0e2-0-0
Degree $4$
Conductor $2624400$
Sign $1$
Analytic cond. $0.653648$
Root an. cond. $0.899158$
Motivic weight $0$
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  − 2·7-s − 2·11-s − 2·17-s − 2·23-s − 25-s + 2·31-s − 2·41-s + 2·49-s + 2·53-s − 2·67-s + 2·71-s + 2·73-s + 4·77-s + 2·101-s + 2·103-s − 2·107-s + 2·113-s + 4·119-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + ⋯
L(s)  = 1  − 2·7-s − 2·11-s − 2·17-s − 2·23-s − 25-s + 2·31-s − 2·41-s + 2·49-s + 2·53-s − 2·67-s + 2·71-s + 2·73-s + 4·77-s + 2·101-s + 2·103-s − 2·107-s + 2·113-s + 4·119-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2624400\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.653648\)
Root analytic conductor: \(0.899158\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1620} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2624400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3089253058\)
\(L(\frac12)\) \(\approx\) \(0.3089253058\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
good7$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
11$C_2$ \( ( 1 + T + T^{2} )^{2} \)
13$C_2^2$ \( 1 + T^{4} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
19$C_2^2$ \( 1 - T^{2} + T^{4} \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
29$C_2^2$ \( 1 - T^{2} + T^{4} \)
31$C_2$ \( ( 1 - T + T^{2} )^{2} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_2$ \( ( 1 + T + T^{2} )^{2} \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
59$C_2^2$ \( 1 - T^{2} + T^{4} \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
71$C_2$ \( ( 1 - T + T^{2} )^{2} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_2^2$ \( 1 + T^{4} \)
89$C_2^2$ \( 1 - T^{2} + T^{4} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−10.19737043187140351711801789303, −9.528469047834042388493293917643, −9.009293671166468499404791861215, −8.480971055099012828025218778166, −8.326325336335636022341127485430, −7.80050111670424135503191945327, −7.43077438352719075513547102893, −6.87441862660598577907019260160, −6.50752955969908720603522920448, −6.13953809572977233238603293969, −5.99615877641375413605742674760, −5.08060148370656358236704242125, −5.07487114245196319272723291505, −4.22266232440364836589613707871, −3.89968880573234909674772514469, −3.36450347605110785160769639157, −2.77690535333401520316094638848, −2.34158363970689025362885140343, −2.02801588623418918675538848601, −0.39798184820396681645332112451, 0.39798184820396681645332112451, 2.02801588623418918675538848601, 2.34158363970689025362885140343, 2.77690535333401520316094638848, 3.36450347605110785160769639157, 3.89968880573234909674772514469, 4.22266232440364836589613707871, 5.07487114245196319272723291505, 5.08060148370656358236704242125, 5.99615877641375413605742674760, 6.13953809572977233238603293969, 6.50752955969908720603522920448, 6.87441862660598577907019260160, 7.43077438352719075513547102893, 7.80050111670424135503191945327, 8.326325336335636022341127485430, 8.480971055099012828025218778166, 9.009293671166468499404791861215, 9.528469047834042388493293917643, 10.19737043187140351711801789303

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