Properties

Label 4-21e2-1.1-c10e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $178.022$
Root an. cond. $3.65273$
Motivic weight $10$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 243·3-s − 1.02e3·4-s + 1.09e4·7-s − 2.48e5·12-s − 1.12e6·13-s − 4.92e6·19-s + 2.65e6·21-s − 9.76e6·25-s − 1.43e7·27-s − 1.11e7·28-s + 4.98e7·31-s + 9.43e7·37-s − 2.72e8·39-s + 4.22e8·43-s − 1.63e8·49-s + 1.14e9·52-s − 1.19e9·57-s + 1.97e8·61-s + 1.07e9·64-s + 1.26e9·67-s − 1.95e9·73-s − 2.37e9·75-s + 5.04e9·76-s − 2.10e9·79-s − 3.48e9·81-s − 2.71e9·84-s − 1.22e10·91-s + ⋯
L(s)  = 1  + 3-s − 4-s + 0.648·7-s − 12-s − 3.01·13-s − 1.98·19-s + 0.648·21-s − 25-s − 27-s − 0.648·28-s + 1.74·31-s + 1.36·37-s − 3.01·39-s + 2.87·43-s − 0.578·49-s + 3.01·52-s − 1.98·57-s + 0.233·61-s + 64-s + 0.933·67-s − 0.944·73-s − 75-s + 1.98·76-s − 0.682·79-s − 81-s − 0.648·84-s − 1.95·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(178.022\)
Root analytic conductor: \(3.65273\)
Motivic weight: \(10\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :5, 5),\ 1)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.066901973\)
\(L(\frac12)\) \(\approx\) \(1.066901973\)
\(L(6)\) 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)$
bad3$C_2$ \( 1 - p^{5} T + p^{10} T^{2} \)
7$C_2$ \( 1 - 10907 T + p^{10} T^{2} \)
good2$C_2$ \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \)
5$C_2$ \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \)
11$C_2$ \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \)
13$C_2$ \( ( 1 + 560257 T + p^{10} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \)
19$C_2$ \( ( 1 + 2024677 T + p^{10} T^{2} )( 1 + 2901574 T + p^{10} T^{2} ) \)
23$C_2$ \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \)
31$C_2$ \( ( 1 - 49326674 T + p^{10} T^{2} )( 1 - 516899 T + p^{10} T^{2} ) \)
37$C_2$ \( ( 1 - 135214586 T + p^{10} T^{2} )( 1 + 40895593 T + p^{10} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \)
43$C_2$ \( ( 1 - 211108739 T + p^{10} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \)
53$C_2$ \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \)
59$C_2$ \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \)
61$C_2$ \( ( 1 - 1551490727 T + p^{10} T^{2} )( 1 + 1354266001 T + p^{10} T^{2} ) \)
67$C_2$ \( ( 1 - 2698325411 T + p^{10} T^{2} )( 1 + 1437442918 T + p^{10} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \)
73$C_2$ \( ( 1 - 2186355743 T + p^{10} T^{2} )( 1 + 4144040686 T + p^{10} T^{2} ) \)
79$C_2$ \( ( 1 - 3959005298 T + p^{10} T^{2} )( 1 + 6059886949 T + p^{10} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \)
89$C_2$ \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \)
97$C_2$ \( ( 1 - 884916482 T + p^{10} T^{2} )^{2} \)
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

−15.97888810443697348681559611756, −14.97873201255367864012863233165, −14.76065536785130689020871524067, −14.28310877244664970252503517109, −13.68115545707350002583155875086, −12.89041580245344538787327752090, −12.32507367826096577054758094033, −11.54366828828261323572962927404, −10.52178958754663682339631532922, −9.530760343666959544330514427914, −9.437747454675062396311481318086, −8.288036093053852830176556787398, −7.967054357579153138303090384502, −7.03934211646579068904780625138, −5.70590921258513929303530905094, −4.44441158647025928907998056370, −4.41139361766337083007247776931, −2.60052370890738563965330904422, −2.22968747529048781515639301354, −0.39492249496548271021533685501, 0.39492249496548271021533685501, 2.22968747529048781515639301354, 2.60052370890738563965330904422, 4.41139361766337083007247776931, 4.44441158647025928907998056370, 5.70590921258513929303530905094, 7.03934211646579068904780625138, 7.967054357579153138303090384502, 8.288036093053852830176556787398, 9.437747454675062396311481318086, 9.530760343666959544330514427914, 10.52178958754663682339631532922, 11.54366828828261323572962927404, 12.32507367826096577054758094033, 12.89041580245344538787327752090, 13.68115545707350002583155875086, 14.28310877244664970252503517109, 14.76065536785130689020871524067, 14.97873201255367864012863233165, 15.97888810443697348681559611756

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