Properties

Label 4-21e2-1.1-c8e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $73.1871$
Root an. cond. $2.92488$
Motivic weight $8$
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  − 81·3-s − 256·4-s + 239·7-s + 2.07e4·12-s − 4.12e4·13-s − 1.00e5·19-s − 1.93e4·21-s − 3.90e5·25-s + 5.31e5·27-s − 6.11e4·28-s − 1.22e6·31-s − 2.96e6·37-s + 3.34e6·39-s − 1.36e7·43-s − 5.70e6·49-s + 1.05e7·52-s + 8.14e6·57-s + 2.38e7·61-s + 1.67e7·64-s − 3.72e7·67-s + 5.52e7·73-s + 3.16e7·75-s + 2.57e7·76-s − 7.48e7·79-s − 4.30e7·81-s + 4.95e6·84-s − 9.86e6·91-s + ⋯
L(s)  = 1  − 3-s − 4-s + 0.0995·7-s + 12-s − 1.44·13-s − 0.771·19-s − 0.0995·21-s − 25-s + 27-s − 0.0995·28-s − 1.32·31-s − 1.58·37-s + 1.44·39-s − 3.99·43-s − 0.990·49-s + 1.44·52-s + 0.771·57-s + 1.72·61-s + 64-s − 1.85·67-s + 1.94·73-s + 75-s + 0.771·76-s − 1.92·79-s − 81-s + 0.0995·84-s − 0.143·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(73.1871\)
Root analytic conductor: \(2.92488\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :4, 4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0007400363183\)
\(L(\frac12)\) \(\approx\) \(0.0007400363183\)
\(L(5)\) 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$C_2$ \( 1 + p^{4} T + p^{8} T^{2} \)
7$C_2$ \( 1 - 239 T + p^{8} T^{2} \)
good2$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \)
5$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \)
11$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \)
13$C_2$ \( ( 1 + 20641 T + p^{8} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \)
19$C_2$ \( ( 1 - 157967 T + p^{8} T^{2} )( 1 + 258526 T + p^{8} T^{2} ) \)
23$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \)
31$C_2$ \( ( 1 - 583439 T + p^{8} T^{2} )( 1 + 1809406 T + p^{8} T^{2} ) \)
37$C_2$ \( ( 1 - 503522 T + p^{8} T^{2} )( 1 + 3468481 T + p^{8} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \)
43$C_2$ \( ( 1 + 6837073 T + p^{8} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \)
53$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \)
59$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \)
61$C_2$ \( ( 1 - 24133919 T + p^{8} T^{2} )( 1 + 307393 T + p^{8} T^{2} ) \)
67$C_2$ \( ( 1 + 5421406 T + p^{8} T^{2} )( 1 + 31874833 T + p^{8} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \)
73$C_2$ \( ( 1 - 39067199 T + p^{8} T^{2} )( 1 - 16169282 T + p^{8} T^{2} ) \)
79$C_2$ \( ( 1 + 18887038 T + p^{8} T^{2} )( 1 + 56007121 T + p^{8} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \)
89$C_2$ \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \)
97$C_2$ \( ( 1 - 176908034 T + p^{8} T^{2} )^{2} \)
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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

−16.98491894259675050641339044290, −16.15264425770588581663712642672, −15.29200932725639350088008541906, −14.58954303639513950325049632208, −14.11315272123204943681388978184, −13.17442605383881753167377198218, −12.76894787036919711784978451655, −11.75080971989746328731517973111, −11.57239418173787500276244334616, −10.36265179348038408059423182347, −9.954580578670223570415162559704, −8.979395043871960646558849491891, −8.313267813178255265750901290622, −7.20321873157896985087515750853, −6.34768266429729243552035725998, −5.13873438574477884754289261089, −4.89786215999606783991360831706, −3.58138282006602117169220338107, −1.93347095642437217861327855793, −0.01416689557175330012724977904, 0.01416689557175330012724977904, 1.93347095642437217861327855793, 3.58138282006602117169220338107, 4.89786215999606783991360831706, 5.13873438574477884754289261089, 6.34768266429729243552035725998, 7.20321873157896985087515750853, 8.313267813178255265750901290622, 8.979395043871960646558849491891, 9.954580578670223570415162559704, 10.36265179348038408059423182347, 11.57239418173787500276244334616, 11.75080971989746328731517973111, 12.76894787036919711784978451655, 13.17442605383881753167377198218, 14.11315272123204943681388978184, 14.58954303639513950325049632208, 15.29200932725639350088008541906, 16.15264425770588581663712642672, 16.98491894259675050641339044290

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