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Properties

Label 4-21e2-1.1-c41e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $49992.7$
Root an. cond. $14.9529$
Motivic weight $41$
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  + 1.04e10·3-s − 2.19e12·4-s + 4.21e17·7-s + 7.29e19·9-s − 2.30e22·12-s + 4.91e26·19-s + 4.40e27·21-s + 4.54e28·25-s + 3.81e29·27-s − 9.26e29·28-s + 1.19e31·31-s − 1.60e32·36-s + 6.47e31·37-s + 3.95e33·43-s + 1.33e35·49-s + 5.14e36·57-s + 1.17e35·61-s + 3.07e37·63-s + 1.06e37·64-s + 5.08e37·67-s + 1.41e37·73-s + 4.75e38·75-s − 1.08e39·76-s + 1.27e39·79-s + 1.33e39·81-s − 9.69e39·84-s + 1.24e41·93-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s + 1.99·7-s + 2·9-s − 1.73·12-s + 3.00·19-s + 3.45·21-s + 25-s + 1.73·27-s − 1.99·28-s + 3.18·31-s − 2·36-s + 0.460·37-s + 1.29·43-s + 2.98·49-s + 5.19·57-s + 0.0296·61-s + 3.99·63-s + 64-s + 1.86·67-s + 0.0895·73-s + 1.73·75-s − 3.00·76-s + 1.60·79-s + 81-s − 3.45·84-s + 5.51·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+41/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(49992.7\)
Root analytic conductor: \(14.9529\)
Motivic weight: \(41\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :41/2, 41/2),\ 1)\)

Particular Values

\(L(21)\) \(\approx\) \(15.96241938\)
\(L(\frac12)\) \(\approx\) \(15.96241938\)
\(L(\frac{43}{2})\) 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$C_2$ \( 1 - p^{21} T + p^{41} T^{2} \)
7$C_2$ \( 1 - 421490503104712000 T + p^{41} T^{2} \)
good2$C_2^2$ \( 1 + p^{41} T^{2} + p^{82} T^{4} \)
5$C_2^2$ \( 1 - p^{41} T^{2} + p^{82} T^{4} \)
11$C_2^2$ \( 1 + p^{41} T^{2} + p^{82} T^{4} \)
13$C_2$ \( ( 1 - \)\(11\!\cdots\!77\)\( T + p^{41} T^{2} )( 1 + \)\(11\!\cdots\!77\)\( T + p^{41} T^{2} ) \)
17$C_2^2$ \( 1 - p^{41} T^{2} + p^{82} T^{4} \)
19$C_2$ \( ( 1 - \)\(32\!\cdots\!49\)\( T + p^{41} T^{2} )( 1 - \)\(16\!\cdots\!32\)\( T + p^{41} T^{2} ) \)
23$C_2^2$ \( 1 + p^{41} T^{2} + p^{82} T^{4} \)
29$C_2$ \( ( 1 - p^{41} T^{2} )^{2} \)
31$C_2$ \( ( 1 - \)\(74\!\cdots\!21\)\( T + p^{41} T^{2} )( 1 - \)\(44\!\cdots\!64\)\( T + p^{41} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(26\!\cdots\!10\)\( T + p^{41} T^{2} )( 1 + \)\(20\!\cdots\!99\)\( T + p^{41} T^{2} ) \)
41$C_2$ \( ( 1 + p^{41} T^{2} )^{2} \)
43$C_2$ \( ( 1 - \)\(19\!\cdots\!45\)\( T + p^{41} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{41} T^{2} + p^{82} T^{4} \)
53$C_2^2$ \( 1 + p^{41} T^{2} + p^{82} T^{4} \)
59$C_2^2$ \( 1 - p^{41} T^{2} + p^{82} T^{4} \)
61$C_2$ \( ( 1 - \)\(40\!\cdots\!13\)\( T + p^{41} T^{2} )( 1 + \)\(39\!\cdots\!61\)\( T + p^{41} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(42\!\cdots\!36\)\( T + p^{41} T^{2} )( 1 - \)\(86\!\cdots\!55\)\( T + p^{41} T^{2} ) \)
71$C_2$ \( ( 1 - p^{41} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(16\!\cdots\!07\)\( T + p^{41} T^{2} )( 1 + \)\(15\!\cdots\!10\)\( T + p^{41} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(14\!\cdots\!87\)\( T + p^{41} T^{2} )( 1 + \)\(18\!\cdots\!04\)\( T + p^{41} T^{2} ) \)
83$C_2$ \( ( 1 + p^{41} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{41} T^{2} + p^{82} T^{4} \)
97$C_2$ \( ( 1 - \)\(89\!\cdots\!06\)\( T + p^{41} T^{2} )( 1 + \)\(89\!\cdots\!06\)\( T + p^{41} T^{2} ) \)
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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

−11.21223140107970787120981735834, −10.35913235546043499243776371673, −9.729557103018931694900227227480, −9.391568685295201280921442882501, −8.792004070059424369435078433936, −8.268597323614005815220727621696, −8.038774479900547269829015890780, −7.46653929108497972248748988285, −7.02468607768345997735193278197, −5.94707336058419198824362251194, −4.95593714403836570156209039707, −4.94168482755603256248106915048, −4.40510669530501219560262078576, −3.72143270098241994231105772719, −3.21721560019215012738844096223, −2.45535495047263783072128051453, −2.31777806108643967231617207371, −1.27117362864011335639735175913, −0.990598765625308840292502345926, −0.793124078707899637314877223205, 0.793124078707899637314877223205, 0.990598765625308840292502345926, 1.27117362864011335639735175913, 2.31777806108643967231617207371, 2.45535495047263783072128051453, 3.21721560019215012738844096223, 3.72143270098241994231105772719, 4.40510669530501219560262078576, 4.94168482755603256248106915048, 4.95593714403836570156209039707, 5.94707336058419198824362251194, 7.02468607768345997735193278197, 7.46653929108497972248748988285, 8.038774479900547269829015890780, 8.268597323614005815220727621696, 8.792004070059424369435078433936, 9.391568685295201280921442882501, 9.729557103018931694900227227480, 10.35913235546043499243776371673, 11.21223140107970787120981735834

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