Properties

Label 4-21e2-1.1-c5e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $11.3438$
Root an. cond. $1.83522$
Motivic weight $5$
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  + 27·3-s − 32·4-s + 211·7-s + 486·9-s − 864·12-s − 279·19-s + 5.69e3·21-s + 3.12e3·25-s + 6.56e3·27-s − 6.75e3·28-s + 1.79e4·31-s − 1.55e4·36-s − 6.66e3·37-s − 4.49e4·43-s + 2.77e4·49-s − 7.53e3·57-s − 7.52e4·61-s + 1.02e5·63-s + 3.27e4·64-s + 3.79e4·67-s − 8.10e4·73-s + 8.43e4·75-s + 8.92e3·76-s − 9.08e4·79-s + 5.90e4·81-s − 1.82e5·84-s + 4.83e5·93-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s + 1.62·7-s + 2·9-s − 1.73·12-s − 0.177·19-s + 2.81·21-s + 25-s + 1.73·27-s − 1.62·28-s + 3.35·31-s − 2·36-s − 0.799·37-s − 3.70·43-s + 1.64·49-s − 0.307·57-s − 2.58·61-s + 3.25·63-s + 64-s + 1.03·67-s − 1.77·73-s + 1.73·75-s + 0.177·76-s − 1.63·79-s + 81-s − 2.81·84-s + 5.80·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(11.3438\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.902171096\)
\(L(\frac12)\) \(\approx\) \(2.902171096\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 - p^{3} T + p^{5} T^{2} \)
7$C_2$ \( 1 - 211 T + p^{5} T^{2} \)
good2$C_2^2$ \( 1 + p^{5} T^{2} + p^{10} T^{4} \)
5$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
11$C_2^2$ \( 1 + p^{5} T^{2} + p^{10} T^{4} \)
13$C_2$ \( ( 1 - 427 T + p^{5} T^{2} )( 1 + 427 T + p^{5} T^{2} ) \)
17$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 1432 T + p^{5} T^{2} )( 1 + 1711 T + p^{5} T^{2} ) \)
23$C_2^2$ \( 1 + p^{5} T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10324 T + p^{5} T^{2} )( 1 - 7601 T + p^{5} T^{2} ) \)
37$C_2$ \( ( 1 - 9889 T + p^{5} T^{2} )( 1 + 16550 T + p^{5} T^{2} ) \)
41$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 22475 T + p^{5} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 + p^{5} T^{2} + p^{10} T^{4} \)
59$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
61$C_2$ \( ( 1 + 18301 T + p^{5} T^{2} )( 1 + 56927 T + p^{5} T^{2} ) \)
67$C_2$ \( ( 1 - 73475 T + p^{5} T^{2} )( 1 + 35536 T + p^{5} T^{2} ) \)
71$C_2$ \( ( 1 - p^{5} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 1450 T + p^{5} T^{2} )( 1 + 79577 T + p^{5} T^{2} ) \)
79$C_2$ \( ( 1 - 9707 T + p^{5} T^{2} )( 1 + 100564 T + p^{5} T^{2} ) \)
83$C_2$ \( ( 1 + p^{5} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{5} T^{2} + p^{10} T^{4} \)
97$C_2$ \( ( 1 - 134386 T + p^{5} T^{2} )( 1 + 134386 T + p^{5} T^{2} ) \)
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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

−17.45106670097062708324859298614, −17.09380091478748034014376654794, −15.88889809540984035800331025499, −15.17665312391879728702916164679, −14.79381128271610411718356820870, −14.17987293565371911678086144675, −13.51148327349410256042706359072, −13.47662108028634921731768651174, −12.25056191461997646022441201145, −11.52544330452200570644093092456, −10.36542466702961823442798597344, −9.792717799753384105591753804808, −8.675767895599749167662661560049, −8.522745617566955560754995627565, −7.906434391186503279923112065439, −6.78400659045666483863415812808, −4.86892729722762588698310935596, −4.43281804449275052807917454414, −2.99607649272218920312010938558, −1.53673781904941817027995831137, 1.53673781904941817027995831137, 2.99607649272218920312010938558, 4.43281804449275052807917454414, 4.86892729722762588698310935596, 6.78400659045666483863415812808, 7.906434391186503279923112065439, 8.522745617566955560754995627565, 8.675767895599749167662661560049, 9.792717799753384105591753804808, 10.36542466702961823442798597344, 11.52544330452200570644093092456, 12.25056191461997646022441201145, 13.47662108028634921731768651174, 13.51148327349410256042706359072, 14.17987293565371911678086144675, 14.79381128271610411718356820870, 15.17665312391879728702916164679, 15.88889809540984035800331025499, 17.09380091478748034014376654794, 17.45106670097062708324859298614

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