Properties

Label 4-21e2-1.1-c27e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $9406.99$
Root an. cond. $9.84833$
Motivic weight $27$
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  + 2.68e8·4-s + 4.67e11·7-s − 7.62e12·9-s + 5.40e16·16-s − 1.49e19·25-s + 1.25e20·28-s − 2.04e21·36-s − 4.71e21·37-s − 4.08e22·43-s + 1.53e23·49-s − 3.56e24·63-s + 9.67e24·64-s − 1.55e25·67-s − 2.56e25·79-s + 5.81e25·81-s − 3.99e27·100-s + 6.70e27·109-s + 2.52e28·112-s + 2.62e28·121-s + 127-s + 131-s + 137-s + 139-s − 4.12e29·144-s − 1.26e30·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2·4-s + 1.82·7-s − 9-s + 3·16-s − 2·25-s + 3.65·28-s − 2·36-s − 3.18·37-s − 3.62·43-s + 2.33·49-s − 1.82·63-s + 4·64-s − 3.45·67-s − 0.618·79-s + 81-s − 4·100-s + 2.09·109-s + 5.47·112-s + 2·121-s − 3·144-s − 6.36·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(9406.99\)
Root analytic conductor: \(9.84833\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(4.651190755\)
\(L(\frac12)\) \(\approx\) \(4.651190755\)
\(L(\frac{29}{2})\) 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)$
bad3$C_2$ \( 1 + p^{27} T^{2} \)
7$C_2$ \( 1 - 467920383820 T + p^{27} T^{2} \)
good2$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
11$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2107227619093030 T + p^{27} T^{2} )( 1 + 2107227619093030 T + p^{27} T^{2} ) \)
17$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 13798777614673016 T + p^{27} T^{2} )( 1 + 13798777614673016 T + p^{27} T^{2} ) \)
23$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
31$C_2$ \( ( 1 - \)\(12\!\cdots\!12\)\( T + p^{27} T^{2} )( 1 + \)\(12\!\cdots\!12\)\( T + p^{27} T^{2} ) \)
37$C_2$ \( ( 1 + \)\(23\!\cdots\!90\)\( T + p^{27} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
43$C_2$ \( ( 1 + \)\(20\!\cdots\!20\)\( T + p^{27} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
53$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
61$C_2$ \( ( 1 - \)\(24\!\cdots\!22\)\( T + p^{27} T^{2} )( 1 + \)\(24\!\cdots\!22\)\( T + p^{27} T^{2} ) \)
67$C_2$ \( ( 1 + \)\(77\!\cdots\!20\)\( T + p^{27} T^{2} )^{2} \)
71$C_2$ \( ( 1 - p^{27} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(26\!\cdots\!90\)\( T + p^{27} T^{2} )( 1 + \)\(26\!\cdots\!90\)\( T + p^{27} T^{2} ) \)
79$C_2$ \( ( 1 + \)\(12\!\cdots\!56\)\( T + p^{27} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{27} T^{2} )^{2} \)
97$C_2$ \( ( 1 - \)\(79\!\cdots\!70\)\( T + p^{27} T^{2} )( 1 + \)\(79\!\cdots\!70\)\( T + p^{27} T^{2} ) \)
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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

−12.23874733454893136068006963820, −11.77578539873234857500530301687, −11.68509462246030437864027379860, −11.03297118142804531556769927621, −10.46872007497625013433709703826, −9.970228839802185168150035925402, −8.501021247232567698251740133826, −8.447450067261896582491769668251, −7.59188568245801507840783088494, −7.20552410411195479258431347769, −6.44336737442072119959139694764, −5.68862987265430938119231365034, −5.37959653283573896230969971547, −4.53769063275819371327239266101, −3.34761987959202979084529110391, −3.18162194803965720185496937340, −1.99140581508136299239510212488, −1.86222731529994638275233278973, −1.52341676774795249843576771581, −0.40251261207247843311409560123, 0.40251261207247843311409560123, 1.52341676774795249843576771581, 1.86222731529994638275233278973, 1.99140581508136299239510212488, 3.18162194803965720185496937340, 3.34761987959202979084529110391, 4.53769063275819371327239266101, 5.37959653283573896230969971547, 5.68862987265430938119231365034, 6.44336737442072119959139694764, 7.20552410411195479258431347769, 7.59188568245801507840783088494, 8.447450067261896582491769668251, 8.501021247232567698251740133826, 9.970228839802185168150035925402, 10.46872007497625013433709703826, 11.03297118142804531556769927621, 11.68509462246030437864027379860, 11.77578539873234857500530301687, 12.23874733454893136068006963820

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