Properties

Label 4-21e2-1.1-c2e2-0-3
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  + 2·2-s − 3·3-s + 4·4-s − 6·5-s − 6·6-s − 7·7-s + 16·8-s + 6·9-s − 12·10-s − 10·11-s − 12·12-s − 14·14-s + 18·15-s + 32·16-s − 12·17-s + 12·18-s + 57·19-s − 24·20-s + 21·21-s − 20·22-s − 40·23-s − 48·24-s − 25-s − 9·27-s − 28·28-s + 32·29-s + 36·30-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 6/5·5-s − 6-s − 7-s + 2·8-s + 2/3·9-s − 6/5·10-s − 0.909·11-s − 12-s − 14-s + 6/5·15-s + 2·16-s − 0.705·17-s + 2/3·18-s + 3·19-s − 6/5·20-s + 21-s − 0.909·22-s − 1.73·23-s − 2·24-s − 0.0399·25-s − 1/3·27-s − 28-s + 1.10·29-s + 6/5·30-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

\(L(\frac{3}{2})\) \(\approx\) \(0.8839576896\)
\(L(\frac12)\) \(\approx\) \(0.8839576896\)
\(L(2)\) 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_1$$\times$$C_2$ \( ( 1 - p T )^{2}( 1 + p T + p^{2} T^{2} ) \)
5$C_2^2$ \( 1 + 6 T + 37 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \)
11$C_2^2$ \( 1 + 10 T - 21 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \)
13$C_2$ \( ( 1 - 23 T + p^{2} T^{2} )( 1 + 23 T + p^{2} T^{2} ) \)
17$C_2^2$ \( 1 + 12 T + 337 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \)
19$C_1$$\times$$C_2$ \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \)
23$C_2^2$ \( 1 + 40 T + 1071 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} \)
29$C_2$ \( ( 1 - 16 T + p^{2} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 9 T + 988 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \)
37$C_2^2$ \( 1 + 5 T - 1344 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2^2$ \( 1 - 2774 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 19 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 90 T + 4909 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 32 T - 1785 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 72 T + 5209 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 36 T + 4153 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 + 59 T - 1008 T^{2} + 59 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2$ \( ( 1 + 26 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 33 T + 5692 T^{2} + 33 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2^2$ \( 1 + 47 T - 4032 T^{2} + 47 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 - 13190 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 204 T + 21793 T^{2} - 204 p^{2} T^{3} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 16466 T^{2} + p^{4} T^{4} \)
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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

−17.98575500164893497617612111575, −17.90683416423905283373161651990, −16.46597840674563552948280763401, −16.28447276859396086641175184325, −15.75797551673013038124459084352, −15.63988802368905518660367202192, −14.34528053835078955911414753800, −13.54432850756280588224298023507, −13.22209945964985216508732192582, −12.27880611898751037709276697879, −11.55461121444636327817579051007, −11.53885667903414972455922494576, −10.13928041292070285369187287890, −10.06348780879576766014782917988, −8.017509227007492090205306035738, −7.47388105315423938103714340800, −6.62750552293455248086161610210, −5.50551512778034746268598153449, −4.61368157550132321146797149827, −3.43300208294605646978948795414, 3.43300208294605646978948795414, 4.61368157550132321146797149827, 5.50551512778034746268598153449, 6.62750552293455248086161610210, 7.47388105315423938103714340800, 8.017509227007492090205306035738, 10.06348780879576766014782917988, 10.13928041292070285369187287890, 11.53885667903414972455922494576, 11.55461121444636327817579051007, 12.27880611898751037709276697879, 13.22209945964985216508732192582, 13.54432850756280588224298023507, 14.34528053835078955911414753800, 15.63988802368905518660367202192, 15.75797551673013038124459084352, 16.28447276859396086641175184325, 16.46597840674563552948280763401, 17.90683416423905283373161651990, 17.98575500164893497617612111575

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