Properties

Label 4-210e2-1.1-c1e2-0-4
Degree $4$
Conductor $44100$
Sign $1$
Analytic cond. $2.81185$
Root an. cond. $1.29493$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 5-s − 6-s − 4·7-s − 8-s + 10-s − 3·11-s + 10·13-s − 4·14-s − 15-s − 16-s − 5·19-s + 4·21-s − 3·22-s + 9·23-s + 24-s + 10·26-s + 27-s − 30-s + 10·31-s + 3·33-s − 4·35-s + 37-s − 5·38-s − 10·39-s − 40-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 0.904·11-s + 2.77·13-s − 1.06·14-s − 0.258·15-s − 1/4·16-s − 1.14·19-s + 0.872·21-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 − 0.676·35-s + 0.164·37-s − 0.811·38-s − 1.60·39-s − 0.158·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2.81185\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{210} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.401141892\)
\(L(\frac12)\) \(\approx\) \(1.401141892\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
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$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.02338460356750465400396986902, −12.31377861508718605690444045591, −11.73606600420673610091317756613, −10.82626068472572192924194291725, −10.82073590091933448591265185424, −10.58958901551304442257682701023, −9.618196050010819659908072570289, −9.092326408704231611440935654637, −8.900696900732510841304835813667, −8.151825859177180696475849251217, −7.47734128277570284904337163165, −6.56254286947573514363381457416, −6.32955907448170344143974490838, −5.83792323122056917118241146102, −5.58825898059678389104892762382, −4.33468279326317223710761222660, −4.22024235165801335142444568945, −2.99224803241864908045648855600, −2.85378636880027908020367740970, −1.02322417706690795548072559386, 1.02322417706690795548072559386, 2.85378636880027908020367740970, 2.99224803241864908045648855600, 4.22024235165801335142444568945, 4.33468279326317223710761222660, 5.58825898059678389104892762382, 5.83792323122056917118241146102, 6.32955907448170344143974490838, 6.56254286947573514363381457416, 7.47734128277570284904337163165, 8.151825859177180696475849251217, 8.900696900732510841304835813667, 9.092326408704231611440935654637, 9.618196050010819659908072570289, 10.58958901551304442257682701023, 10.82073590091933448591265185424, 10.82626068472572192924194291725, 11.73606600420673610091317756613, 12.31377861508718605690444045591, 13.02338460356750465400396986902

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