Properties

Label 4-21e2-1.1-c2e2-0-1
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $0.327422$
Root an. cond. $0.756444$
Motivic weight $2$
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-s + 3·3-s + 4·4-s − 9·5-s − 3·6-s − 7·7-s − 11·8-s + 6·9-s + 9·10-s + 11·11-s + 12·12-s + 7·14-s − 27·15-s + 11·16-s + 42·17-s − 6·18-s − 6·19-s − 36·20-s − 21·21-s − 11·22-s − 28·23-s − 33·24-s + 29·25-s + 9·27-s − 28·28-s + 50·29-s + 27·30-s + ⋯
L(s)  = 1  − 1/2·2-s + 3-s + 4-s − 9/5·5-s − 1/2·6-s − 7-s − 1.37·8-s + 2/3·9-s + 9/10·10-s + 11-s + 12-s + 1/2·14-s − 9/5·15-s + 0.687·16-s + 2.47·17-s − 1/3·18-s − 0.315·19-s − 9/5·20-s − 21-s − 1/2·22-s − 1.21·23-s − 1.37·24-s + 1.15·25-s + 1/3·27-s − 28-s + 1.72·29-s + 9/10·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.327422\)
Root analytic conductor: \(0.756444\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7078107880\)
\(L(\frac12)\) \(\approx\) \(0.7078107880\)
\(L(2)\) 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$C_2$ \( 1 - p T + p T^{2} \)
7$C_2$ \( 1 + p T + p^{2} T^{2} \)
good2$C_2^2$ \( 1 + T - 3 T^{2} + p^{2} T^{3} + p^{4} T^{4} \)
5$C_2^2$ \( 1 + 9 T + 52 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \)
11$C_1$$\times$$C_2$ \( ( 1 - p T )^{2}( 1 + p T + p^{2} T^{2} ) \)
13$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
17$C_2^2$ \( 1 - 42 T + 877 T^{2} - 42 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2^2$ \( 1 + 6 T + 373 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
23$C_2^2$ \( 1 + 28 T + 255 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 25 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 57 T + 2044 T^{2} + 57 p^{2} T^{3} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 58 T + 1995 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 3350 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 132 T + 8017 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} \)
53$C_2^2$ \( 1 + 31 T - 1848 T^{2} + 31 p^{2} T^{3} + p^{4} T^{4} \)
59$C_2^2$ \( 1 + 15 T + 3556 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 24 T + 3913 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 52 T - 1785 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 - 64 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 12 T + 5377 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2^2$ \( 1 + 17 T - 5952 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 10895 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 + 138 T + 14269 T^{2} + 138 p^{2} T^{3} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 10391 T^{2} + p^{4} T^{4} \)
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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

−18.57487378858281785048697742474, −17.87713402532611485950052584663, −16.59476181623643885669766982873, −16.32490042475320846356153975606, −15.85860717688117591062652482968, −15.25478531882660163329757668504, −14.46196494531299827678109479040, −14.34633160381525503452410563419, −12.70873457819104802394232496202, −12.34463758149409219845136168174, −11.75592968208496928270867203428, −11.11304553140246309728494652812, −9.820437134345424814553307718775, −9.494130765849989756054432642969, −8.269149256054203646871253978662, −7.910717366185993546844553523571, −7.00377100269439949615422199973, −6.11761929321551634880283495148, −3.80768540697067178356210859712, −3.17648036595521212323167511635, 3.17648036595521212323167511635, 3.80768540697067178356210859712, 6.11761929321551634880283495148, 7.00377100269439949615422199973, 7.910717366185993546844553523571, 8.269149256054203646871253978662, 9.494130765849989756054432642969, 9.820437134345424814553307718775, 11.11304553140246309728494652812, 11.75592968208496928270867203428, 12.34463758149409219845136168174, 12.70873457819104802394232496202, 14.34633160381525503452410563419, 14.46196494531299827678109479040, 15.25478531882660163329757668504, 15.85860717688117591062652482968, 16.32490042475320846356153975606, 16.59476181623643885669766982873, 17.87713402532611485950052584663, 18.57487378858281785048697742474

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