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Properties

Label 4-21e2-1.1-c1e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $0.0281185$
Root an. cond. $0.409494$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s + 2·5-s + 2·6-s − 5·7-s − 4·8-s − 4·10-s + 2·11-s − 2·12-s + 2·13-s + 10·14-s − 2·15-s + 8·16-s − 19-s + 4·20-s + 5·21-s − 4·22-s + 4·24-s + 5·25-s − 4·26-s + 27-s − 10·28-s + 8·29-s + 4·30-s − 9·31-s − 8·32-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.894·5-s + 0.816·6-s − 1.88·7-s − 1.41·8-s − 1.26·10-s + 0.603·11-s − 0.577·12-s + 0.554·13-s + 2.67·14-s − 0.516·15-s + 2·16-s − 0.229·19-s + 0.894·20-s + 1.09·21-s − 0.852·22-s + 0.816·24-s + 25-s − 0.784·26-s + 0.192·27-s − 1.88·28-s + 1.48·29-s + 0.730·30-s − 1.61·31-s − 1.41·32-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.0281185\)
Root analytic conductor: \(0.409494\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1886313601\)
\(L(\frac12)\) \(\approx\) \(0.1886313601\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + 5 T + p T^{2} \)
good2$C_2^2$ \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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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

−18.35162599690320300427413645604, −18.12129056260894093908436548569, −17.20622136474437902816441882031, −17.04031378858136437073408927109, −16.32959379130278819697921305148, −15.79761123923961004510355109668, −15.07414260816926352661587808986, −14.09592991053196328885893654920, −13.34323482627396490642499025152, −12.46535281282447583047007665638, −12.15250100734095017521574237444, −10.97701824034660861287431573674, −10.22619556082021660311570594860, −9.745945150935527986080392896027, −9.036766158069173391425947412212, −8.601280514767607011404874617841, −6.94429713931444545466421079467, −6.38832905030081022881130156037, −5.69031773444673653876183633556, −3.25829500936861479080741427724, 3.25829500936861479080741427724, 5.69031773444673653876183633556, 6.38832905030081022881130156037, 6.94429713931444545466421079467, 8.601280514767607011404874617841, 9.036766158069173391425947412212, 9.745945150935527986080392896027, 10.22619556082021660311570594860, 10.97701824034660861287431573674, 12.15250100734095017521574237444, 12.46535281282447583047007665638, 13.34323482627396490642499025152, 14.09592991053196328885893654920, 15.07414260816926352661587808986, 15.79761123923961004510355109668, 16.32959379130278819697921305148, 17.04031378858136437073408927109, 17.20622136474437902816441882031, 18.12129056260894093908436548569, 18.35162599690320300427413645604

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