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Properties

Label 4-21e2-1.1-c28e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $10879.2$
Root an. cond. $10.2129$
Motivic weight $28$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.78e6·3-s − 2.68e8·4-s + 7.78e11·7-s + 1.28e15·12-s + 1.37e16·13-s − 9.03e17·19-s − 3.72e18·21-s − 3.72e19·25-s + 1.09e20·27-s − 2.08e20·28-s − 1.48e21·31-s + 1.20e22·37-s − 6.57e22·39-s − 1.28e22·43-s + 1.45e23·49-s − 3.69e24·52-s + 4.32e24·57-s − 1.88e25·61-s + 1.93e25·64-s + 1.53e25·67-s − 1.78e26·73-s + 1.78e26·75-s + 2.42e26·76-s + 4.13e26·79-s − 5.23e26·81-s + 9.99e26·84-s + 1.07e28·91-s + ⋯
L(s)  = 1  − 3-s − 4-s + 1.14·7-s + 12-s + 3.49·13-s − 1.13·19-s − 1.14·21-s − 25-s + 27-s − 1.14·28-s − 1.96·31-s + 1.33·37-s − 3.49·39-s − 0.174·43-s + 0.316·49-s − 3.49·52-s + 1.13·57-s − 1.91·61-s + 64-s + 0.417·67-s − 1.46·73-s + 75-s + 1.13·76-s + 1.12·79-s − 81-s + 1.14·84-s + 4.00·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(29-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+14)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(10879.2\)
Root analytic conductor: \(10.2129\)
Motivic weight: \(28\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :14, 14),\ 1)\)

Particular Values

\(L(\frac{29}{2})\) \(\approx\) \(0.2573241458\)
\(L(\frac12)\) \(\approx\) \(0.2573241458\)
\(L(15)\) 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$C_2$ \( 1 + p^{14} T + p^{28} T^{2} \)
7$C_2$ \( 1 - 778317387191 T + p^{28} T^{2} \)
good2$C_2$ \( ( 1 - p^{14} T + p^{28} T^{2} )( 1 + p^{14} T + p^{28} T^{2} ) \)
5$C_2$ \( ( 1 - p^{14} T + p^{28} T^{2} )( 1 + p^{14} T + p^{28} T^{2} ) \)
11$C_2$ \( ( 1 - p^{14} T + p^{28} T^{2} )( 1 + p^{14} T + p^{28} T^{2} ) \)
13$C_2$ \( ( 1 - 6878010332269631 T + p^{28} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{14} T + p^{28} T^{2} )( 1 + p^{14} T + p^{28} T^{2} ) \)
19$C_2$ \( ( 1 - 689731287938390834 T + p^{28} T^{2} )( 1 + 1593239832506583433 T + p^{28} T^{2} ) \)
23$C_2$ \( ( 1 - p^{14} T + p^{28} T^{2} )( 1 + p^{14} T + p^{28} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{14} T )^{2}( 1 + p^{14} T )^{2} \)
31$C_2$ \( ( 1 + \)\(50\!\cdots\!81\)\( T + p^{28} T^{2} )( 1 + \)\(98\!\cdots\!86\)\( T + p^{28} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(17\!\cdots\!91\)\( T + p^{28} T^{2} )( 1 + \)\(55\!\cdots\!22\)\( T + p^{28} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{14} T )^{2}( 1 + p^{14} T )^{2} \)
43$C_2$ \( ( 1 + \)\(64\!\cdots\!77\)\( T + p^{28} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{14} T + p^{28} T^{2} )( 1 + p^{14} T + p^{28} T^{2} ) \)
53$C_2$ \( ( 1 - p^{14} T + p^{28} T^{2} )( 1 + p^{14} T + p^{28} T^{2} ) \)
59$C_2$ \( ( 1 - p^{14} T + p^{28} T^{2} )( 1 + p^{14} T + p^{28} T^{2} ) \)
61$C_2$ \( ( 1 + \)\(43\!\cdots\!01\)\( T + p^{28} T^{2} )( 1 + \)\(14\!\cdots\!93\)\( T + p^{28} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(69\!\cdots\!86\)\( T + p^{28} T^{2} )( 1 + \)\(54\!\cdots\!17\)\( T + p^{28} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{14} T )^{2}( 1 + p^{14} T )^{2} \)
73$C_2$ \( ( 1 - \)\(54\!\cdots\!18\)\( T + p^{28} T^{2} )( 1 + \)\(23\!\cdots\!69\)\( T + p^{28} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(73\!\cdots\!62\)\( T + p^{28} T^{2} )( 1 + \)\(32\!\cdots\!41\)\( T + p^{28} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{14} T )^{2}( 1 + p^{14} T )^{2} \)
89$C_2$ \( ( 1 - p^{14} T + p^{28} T^{2} )( 1 + p^{14} T + p^{28} T^{2} ) \)
97$C_2$ \( ( 1 + \)\(12\!\cdots\!54\)\( T + p^{28} T^{2} )^{2} \)
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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

−13.00843820405041133151491727064, −11.54370486128746280167623697518, −11.47243957370491037938932419505, −10.72613567719885376904870517480, −10.61725803123886756213023121649, −9.197245163761349289297828314542, −8.960817714515674781069939160387, −8.229252514827666310030946348831, −7.961301172217900841832423389435, −6.70864255215723153107750929289, −6.08919488974275390680907888364, −5.73745651784384568198677077457, −5.11194860643855725247459759428, −4.19035164425144225049422445694, −4.08574377230321824985297617897, −3.29601253362586455047019551353, −2.12118093713911370843700142234, −1.29948465130286345742570753535, −1.17203519139096739324049184661, −0.13383214249565080815693335820, 0.13383214249565080815693335820, 1.17203519139096739324049184661, 1.29948465130286345742570753535, 2.12118093713911370843700142234, 3.29601253362586455047019551353, 4.08574377230321824985297617897, 4.19035164425144225049422445694, 5.11194860643855725247459759428, 5.73745651784384568198677077457, 6.08919488974275390680907888364, 6.70864255215723153107750929289, 7.961301172217900841832423389435, 8.229252514827666310030946348831, 8.960817714515674781069939160387, 9.197245163761349289297828314542, 10.61725803123886756213023121649, 10.72613567719885376904870517480, 11.47243957370491037938932419505, 11.54370486128746280167623697518, 13.00843820405041133151491727064

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