Properties

Label 4-21e2-1.1-c7e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $43.0347$
Root an. cond. $2.56126$
Motivic weight $7$
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 − 128·4-s − 1.76e3·7-s + 4.37e3·9-s − 1.03e4·12-s + 1.00e5·19-s − 1.42e5·21-s + 7.81e4·25-s + 1.77e5·27-s + 2.25e5·28-s − 2.64e4·31-s − 5.59e5·36-s + 3.35e5·37-s + 8.18e5·43-s + 2.28e6·49-s + 8.14e6·57-s − 4.61e5·61-s − 7.71e6·63-s + 2.09e6·64-s − 4.44e6·67-s − 1.13e7·73-s + 6.32e6·75-s − 1.28e7·76-s + 4.51e6·79-s + 4.78e6·81-s + 1.82e7·84-s − 2.14e6·93-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s − 1.94·7-s + 2·9-s − 1.73·12-s + 3.36·19-s − 3.36·21-s + 25-s + 1.73·27-s + 1.94·28-s − 0.159·31-s − 2·36-s + 1.08·37-s + 1.57·43-s + 2.77·49-s + 5.82·57-s − 0.260·61-s − 3.88·63-s + 64-s − 1.80·67-s − 3.40·73-s + 1.73·75-s − 3.36·76-s + 1.03·79-s + 81-s + 3.36·84-s − 0.276·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(43.0347\)
Root analytic conductor: \(2.56126\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.526729267\)
\(L(\frac12)\) \(\approx\) \(2.526729267\)
\(L(\frac{9}{2})\) 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$C_2$ \( 1 - p^{4} T + p^{7} T^{2} \)
7$C_2$ \( 1 + 1763 T + p^{7} T^{2} \)
good2$C_2^2$ \( 1 + p^{7} T^{2} + p^{14} T^{4} \)
5$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
11$C_2^2$ \( 1 + p^{7} T^{2} + p^{14} T^{4} \)
13$C_2$ \( ( 1 - 2009 T + p^{7} T^{2} )( 1 + 2009 T + p^{7} T^{2} ) \)
17$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
19$C_2$ \( ( 1 - 57448 T + p^{7} T^{2} )( 1 - 43091 T + p^{7} T^{2} ) \)
23$C_2^2$ \( 1 + p^{7} T^{2} + p^{14} T^{4} \)
29$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 152471 T + p^{7} T^{2} )( 1 + 178916 T + p^{7} T^{2} ) \)
37$C_2$ \( ( 1 - 615373 T + p^{7} T^{2} )( 1 + 279710 T + p^{7} T^{2} ) \)
41$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 409495 T + p^{7} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
53$C_2^2$ \( 1 + p^{7} T^{2} + p^{14} T^{4} \)
59$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
61$C_2$ \( ( 1 - 1537199 T + p^{7} T^{2} )( 1 + 1998347 T + p^{7} T^{2} ) \)
67$C_2$ \( ( 1 + 385072 T + p^{7} T^{2} )( 1 + 4058455 T + p^{7} T^{2} ) \)
71$C_2$ \( ( 1 - p^{7} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 5038001 T + p^{7} T^{2} )( 1 + 6274810 T + p^{7} T^{2} ) \)
79$C_2$ \( ( 1 - 8763044 T + p^{7} T^{2} )( 1 + 4245427 T + p^{7} T^{2} ) \)
83$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
97$C_2$ \( ( 1 - 12245198 T + p^{7} T^{2} )( 1 + 12245198 T + p^{7} T^{2} ) \)
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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

−16.60287563790474235831036784836, −15.91019834058981815536799198426, −15.84587729529559055110518676044, −14.75088576243766584298235844614, −14.16148773160337675803143766646, −13.56468422893883351378953679894, −13.29016365468036665283568083573, −12.65286444095662741402176102182, −11.79874764689413849130128738174, −10.31771637685868452722618058280, −9.623078507633785008622775172761, −9.289107380849104356005087199310, −8.855315627395578771944507194949, −7.60970864584907729230283543163, −7.13851234781898857230060019209, −5.76586440869246166285909646254, −4.39426827804925410853316034942, −3.29542571803094258523090078040, −2.90567643428056068720684602728, −0.864707980054412244225787254744, 0.864707980054412244225787254744, 2.90567643428056068720684602728, 3.29542571803094258523090078040, 4.39426827804925410853316034942, 5.76586440869246166285909646254, 7.13851234781898857230060019209, 7.60970864584907729230283543163, 8.855315627395578771944507194949, 9.289107380849104356005087199310, 9.623078507633785008622775172761, 10.31771637685868452722618058280, 11.79874764689413849130128738174, 12.65286444095662741402176102182, 13.29016365468036665283568083573, 13.56468422893883351378953679894, 14.16148773160337675803143766646, 14.75088576243766584298235844614, 15.84587729529559055110518676044, 15.91019834058981815536799198426, 16.60287563790474235831036784836

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