Properties

Label 4-21e2-1.1-c9e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $116.980$
Root an. cond. $3.28873$
Motivic weight $9$
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  + 30·2-s − 162·3-s − 4·4-s + 1.12e3·5-s − 4.86e3·6-s + 4.80e3·7-s − 1.18e4·8-s + 1.96e4·9-s + 3.38e4·10-s + 7.32e4·11-s + 648·12-s + 1.41e5·13-s + 1.44e5·14-s − 1.82e5·15-s − 1.54e5·16-s − 1.01e5·17-s + 5.90e5·18-s + 4.81e5·19-s − 4.51e3·20-s − 7.77e5·21-s + 2.19e6·22-s − 9.82e5·23-s + 1.92e6·24-s − 1.26e6·25-s + 4.23e6·26-s − 2.12e6·27-s − 1.92e4·28-s + ⋯
L(s)  = 1  + 1.32·2-s − 1.15·3-s − 0.00781·4-s + 0.807·5-s − 1.53·6-s + 0.755·7-s − 1.02·8-s + 9-s + 1.07·10-s + 1.50·11-s + 0.00902·12-s + 1.37·13-s + 1.00·14-s − 0.931·15-s − 0.587·16-s − 0.295·17-s + 1.32·18-s + 0.848·19-s − 0.00630·20-s − 0.872·21-s + 2.00·22-s − 0.731·23-s + 1.18·24-s − 0.645·25-s + 1.81·26-s − 0.769·27-s − 0.00590·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(116.980\)
Root analytic conductor: \(3.28873\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :9/2, 9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(3.825699349\)
\(L(\frac12)\) \(\approx\) \(3.825699349\)
\(L(\frac{11}{2})\) 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$C_1$ \( ( 1 + p^{4} T )^{2} \)
7$C_1$ \( ( 1 - p^{4} T )^{2} \)
good2$D_{4}$ \( 1 - 15 p T + 113 p^{3} T^{2} - 15 p^{10} T^{3} + p^{18} T^{4} \)
5$D_{4}$ \( 1 - 1128 T + 2533846 T^{2} - 1128 p^{9} T^{3} + p^{18} T^{4} \)
11$D_{4}$ \( 1 - 73284 T + 5040255766 T^{2} - 73284 p^{9} T^{3} + p^{18} T^{4} \)
13$D_{4}$ \( 1 - 141100 T + 26090120766 T^{2} - 141100 p^{9} T^{3} + p^{18} T^{4} \)
17$D_{4}$ \( 1 + 101784 T + 163890278158 T^{2} + 101784 p^{9} T^{3} + p^{18} T^{4} \)
19$D_{4}$ \( 1 - 481744 T + 700509251862 T^{2} - 481744 p^{9} T^{3} + p^{18} T^{4} \)
23$D_{4}$ \( 1 + 982212 T + 684471671662 T^{2} + 982212 p^{9} T^{3} + p^{18} T^{4} \)
29$D_{4}$ \( 1 + 2550924 T + 23659017143182 T^{2} + 2550924 p^{9} T^{3} + p^{18} T^{4} \)
31$D_{4}$ \( 1 + 4935848 T + 45703928191038 T^{2} + 4935848 p^{9} T^{3} + p^{18} T^{4} \)
37$D_{4}$ \( 1 + 16256516 T + 258501173397198 T^{2} + 16256516 p^{9} T^{3} + p^{18} T^{4} \)
41$D_{4}$ \( 1 - 48707856 T + 1234544246122606 T^{2} - 48707856 p^{9} T^{3} + p^{18} T^{4} \)
43$D_{4}$ \( 1 - 7989640 T + 1011320095256166 T^{2} - 7989640 p^{9} T^{3} + p^{18} T^{4} \)
47$D_{4}$ \( 1 - 85572408 T + 3865344002776030 T^{2} - 85572408 p^{9} T^{3} + p^{18} T^{4} \)
53$D_{4}$ \( 1 - 26565324 T + 1785888321799630 T^{2} - 26565324 p^{9} T^{3} + p^{18} T^{4} \)
59$D_{4}$ \( 1 - 115200960 T + 19706779116773158 T^{2} - 115200960 p^{9} T^{3} + p^{18} T^{4} \)
61$D_{4}$ \( 1 + 142820204 T + 14471165594380686 T^{2} + 142820204 p^{9} T^{3} + p^{18} T^{4} \)
67$D_{4}$ \( 1 - 27521392 T + 42449494594232310 T^{2} - 27521392 p^{9} T^{3} + p^{18} T^{4} \)
71$D_{4}$ \( 1 - 38070180 T - 34252321852949618 T^{2} - 38070180 p^{9} T^{3} + p^{18} T^{4} \)
73$D_{4}$ \( 1 + 2095316 T + 100469358493934310 T^{2} + 2095316 p^{9} T^{3} + p^{18} T^{4} \)
79$D_{4}$ \( 1 - 435097048 T + 140284960617569694 T^{2} - 435097048 p^{9} T^{3} + p^{18} T^{4} \)
83$D_{4}$ \( 1 - 264288744 T - 19361594307718730 T^{2} - 264288744 p^{9} T^{3} + p^{18} T^{4} \)
89$D_{4}$ \( 1 - 642673776 T + 754914610596232462 T^{2} - 642673776 p^{9} T^{3} + p^{18} T^{4} \)
97$D_{4}$ \( 1 - 345361228 T + 1029300318136190550 T^{2} - 345361228 p^{9} T^{3} + p^{18} T^{4} \)
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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

−16.21404068085000813057131367033, −15.77006470555963330795549373351, −14.72358835204785960791530225231, −14.24531205305790151143738735743, −13.64026059167467913783446421638, −13.36488871448583770343799387143, −12.29479911786376944326282829969, −12.02031472836377679888204302437, −11.10517144142646406000360805288, −10.64353060459556204223785139281, −9.316233096593683073329850306473, −9.073005853398081980632077338622, −7.61130721668408185089674501820, −6.49014808920787810981015330795, −5.64603184960363974815886968930, −5.48417042246342562458304416093, −4.06720656537130165932443297724, −3.96576619694277359012179679438, −1.81876665678673780041042321127, −0.858568094559562882534316649011, 0.858568094559562882534316649011, 1.81876665678673780041042321127, 3.96576619694277359012179679438, 4.06720656537130165932443297724, 5.48417042246342562458304416093, 5.64603184960363974815886968930, 6.49014808920787810981015330795, 7.61130721668408185089674501820, 9.073005853398081980632077338622, 9.316233096593683073329850306473, 10.64353060459556204223785139281, 11.10517144142646406000360805288, 12.02031472836377679888204302437, 12.29479911786376944326282829969, 13.36488871448583770343799387143, 13.64026059167467913783446421638, 14.24531205305790151143738735743, 14.72358835204785960791530225231, 15.77006470555963330795549373351, 16.21404068085000813057131367033

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