Properties

Label 4-462e2-1.1-c1e2-0-15
Degree $4$
Conductor $213444$
Sign $1$
Analytic cond. $13.6093$
Root an. cond. $1.92070$
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 + 6-s + 4·7-s − 8-s + 11-s + 8·13-s + 4·14-s − 16-s + 17-s + 3·19-s + 4·21-s + 22-s + 23-s − 24-s + 5·25-s + 8·26-s − 27-s − 2·29-s − 6·31-s + 33-s + 34-s + 3·37-s + 3·38-s + 8·39-s − 12·41-s + 4·42-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 0.301·11-s + 2.21·13-s + 1.06·14-s − 1/4·16-s + 0.242·17-s + 0.688·19-s + 0.872·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s + 25-s + 1.56·26-s − 0.192·27-s − 0.371·29-s − 1.07·31-s + 0.174·33-s + 0.171·34-s + 0.493·37-s + 0.486·38-s + 1.28·39-s − 1.87·41-s + 0.617·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(13.6093\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{462} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 213444,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.896296619\)
\(L(\frac12)\) \(\approx\) \(3.896296619\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
11$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 12 T + 71 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−11.18739641860029402026088233978, −11.17771582984605063912808312856, −10.28900000896782880103717838007, −10.22562872242445896207014746238, −9.175670119333603190562741416483, −8.876802409750981891691028317180, −8.679677599137892028760317313119, −8.233858461217780113791476774895, −7.49786838631527557145817345188, −7.38473853408885405857455393289, −6.50394873964644615370761367126, −6.03396418144715189455885145866, −5.48608931412836651163902807347, −5.13112054639530688209121993443, −4.28786695802210391199229024100, −4.13199292402678315299490993059, −3.22274983635293859585433773869, −3.00935733405640523267731201922, −1.66011252726681229380229248115, −1.36724753470873787478607688365, 1.36724753470873787478607688365, 1.66011252726681229380229248115, 3.00935733405640523267731201922, 3.22274983635293859585433773869, 4.13199292402678315299490993059, 4.28786695802210391199229024100, 5.13112054639530688209121993443, 5.48608931412836651163902807347, 6.03396418144715189455885145866, 6.50394873964644615370761367126, 7.38473853408885405857455393289, 7.49786838631527557145817345188, 8.233858461217780113791476774895, 8.679677599137892028760317313119, 8.876802409750981891691028317180, 9.175670119333603190562741416483, 10.22562872242445896207014746238, 10.28900000896782880103717838007, 11.17771582984605063912808312856, 11.18739641860029402026088233978

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