Properties

Label 4-21e2-1.1-c4e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $4.71223$
Root an. cond. $1.47335$
Motivic weight $4$
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  − 9·3-s − 16·4-s + 71·7-s + 144·12-s + 382·13-s + 601·19-s − 639·21-s − 625·25-s + 729·27-s − 1.13e3·28-s + 1.75e3·31-s − 2.59e3·37-s − 3.43e3·39-s + 46·43-s + 2.64e3·49-s − 6.11e3·52-s − 5.40e3·57-s + 1.96e3·61-s + 4.09e3·64-s + 8.80e3·67-s + 1.24e3·73-s + 5.62e3·75-s − 9.61e3·76-s + 1.23e4·79-s − 6.56e3·81-s + 1.02e4·84-s + 2.71e4·91-s + ⋯
L(s)  = 1  − 3-s − 4-s + 1.44·7-s + 12-s + 2.26·13-s + 1.66·19-s − 1.44·21-s − 25-s + 27-s − 1.44·28-s + 1.82·31-s − 1.89·37-s − 2.26·39-s + 0.0248·43-s + 1.09·49-s − 2.26·52-s − 1.66·57-s + 0.528·61-s + 64-s + 1.96·67-s + 0.234·73-s + 75-s − 1.66·76-s + 1.98·79-s − 81-s + 1.44·84-s + 3.27·91-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4.71223\)
Root analytic conductor: \(1.47335\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.007791746\)
\(L(\frac12)\) \(\approx\) \(1.007791746\)
\(L(3)\) 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$C_2$ \( 1 + p^{2} T + p^{4} T^{2} \)
7$C_2$ \( 1 - 71 T + p^{4} T^{2} \)
good2$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \)
5$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \)
11$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \)
13$C_2$ \( ( 1 - 191 T + p^{4} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \)
19$C_2$ \( ( 1 - 647 T + p^{4} T^{2} )( 1 + 46 T + p^{4} T^{2} ) \)
23$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
31$C_2$ \( ( 1 - 1559 T + p^{4} T^{2} )( 1 - 194 T + p^{4} T^{2} ) \)
37$C_2$ \( ( 1 + 529 T + p^{4} T^{2} )( 1 + 2062 T + p^{4} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
43$C_2$ \( ( 1 - 23 T + p^{4} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \)
53$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \)
59$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \)
61$C_2$ \( ( 1 - 7199 T + p^{4} T^{2} )( 1 + 5233 T + p^{4} T^{2} ) \)
67$C_2$ \( ( 1 - 5906 T + p^{4} T^{2} )( 1 - 2903 T + p^{4} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
73$C_2$ \( ( 1 - 9791 T + p^{4} T^{2} )( 1 + 8542 T + p^{4} T^{2} ) \)
79$C_2$ \( ( 1 - 7682 T + p^{4} T^{2} )( 1 - 4679 T + p^{4} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
89$C_2$ \( ( 1 - p^{2} T + p^{4} T^{2} )( 1 + p^{2} T + p^{4} T^{2} ) \)
97$C_2$ \( ( 1 + 18814 T + p^{4} T^{2} )^{2} \)
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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

−17.66060956208785914369390168311, −17.50039957039732988858801580810, −16.54033783760019777619329157182, −15.78206950005915816353246692068, −15.43041620240432757634491334678, −14.05314343381431385109786083765, −13.95995845709668567419156554226, −13.43413179410918312099670764471, −12.26054312942087646333414344231, −11.56977155817409141777589373308, −11.21909595757549120959511927269, −10.45298405814438869166039595380, −9.428429490297320892806909316934, −8.450007410165770593327849381142, −8.134032882213174129328589700882, −6.66159774954992762152960013723, −5.58141662467568535731688105827, −4.99130051023010146226522750460, −3.82602547844482866508531298386, −1.09447029849217569637223041483, 1.09447029849217569637223041483, 3.82602547844482866508531298386, 4.99130051023010146226522750460, 5.58141662467568535731688105827, 6.66159774954992762152960013723, 8.134032882213174129328589700882, 8.450007410165770593327849381142, 9.428429490297320892806909316934, 10.45298405814438869166039595380, 11.21909595757549120959511927269, 11.56977155817409141777589373308, 12.26054312942087646333414344231, 13.43413179410918312099670764471, 13.95995845709668567419156554226, 14.05314343381431385109786083765, 15.43041620240432757634491334678, 15.78206950005915816353246692068, 16.54033783760019777619329157182, 17.50039957039732988858801580810, 17.66060956208785914369390168311

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