Properties

Label 4-21e2-1.1-c31e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $16343.5$
Root an. cond. $11.3067$
Motivic weight $31$
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  + 4.30e7·3-s − 2.14e9·4-s + 9.53e12·7-s + 1.23e15·9-s − 9.24e16·12-s − 1.85e20·19-s + 4.10e20·21-s + 4.65e21·25-s + 2.65e22·27-s − 2.04e22·28-s + 2.74e23·31-s − 2.65e24·36-s + 2.25e24·37-s − 3.55e25·43-s − 6.68e25·49-s − 8.00e27·57-s + 1.62e28·61-s + 1.17e28·63-s + 9.90e27·64-s − 3.02e28·67-s + 2.29e29·73-s + 2.00e29·75-s + 3.99e29·76-s − 1.50e29·79-s + 3.81e29·81-s − 8.81e29·84-s + 1.18e31·93-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s + 0.759·7-s + 2·9-s − 1.73·12-s − 2.80·19-s + 1.31·21-s + 25-s + 1.73·27-s − 0.759·28-s + 2.10·31-s − 2·36-s + 1.11·37-s − 1.70·43-s − 0.423·49-s − 4.86·57-s + 3.44·61-s + 1.51·63-s + 64-s − 1.50·67-s + 3.01·73-s + 1.73·75-s + 2.80·76-s − 0.580·79-s + 81-s − 1.31·84-s + 3.64·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(16343.5\)
Root analytic conductor: \(11.3067\)
Motivic weight: \(31\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :31/2, 31/2),\ 1)\)

Particular Values

\(L(16)\) \(\approx\) \(4.767387633\)
\(L(\frac12)\) \(\approx\) \(4.767387633\)
\(L(\frac{33}{2})\) 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$C_2$ \( 1 - p^{16} T + p^{31} T^{2} \)
7$C_2$ \( 1 - 9534530086957 T + p^{31} T^{2} \)
good2$C_2^2$ \( 1 + p^{31} T^{2} + p^{62} T^{4} \)
5$C_2^2$ \( 1 - p^{31} T^{2} + p^{62} T^{4} \)
11$C_2^2$ \( 1 + p^{31} T^{2} + p^{62} T^{4} \)
13$C_2$ \( ( 1 - 121177137759872551 T + p^{31} T^{2} )( 1 + 121177137759872551 T + p^{31} T^{2} ) \)
17$C_2^2$ \( 1 - p^{31} T^{2} + p^{62} T^{4} \)
19$C_2$ \( ( 1 + 54230691858525415549 T + p^{31} T^{2} )( 1 + \)\(13\!\cdots\!92\)\( T + p^{31} T^{2} ) \)
23$C_2^2$ \( 1 + p^{31} T^{2} + p^{62} T^{4} \)
29$C_2$ \( ( 1 - p^{31} T^{2} )^{2} \)
31$C_2$ \( ( 1 - \)\(24\!\cdots\!04\)\( T + p^{31} T^{2} )( 1 - \)\(33\!\cdots\!11\)\( T + p^{31} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(40\!\cdots\!70\)\( T + p^{31} T^{2} )( 1 + \)\(17\!\cdots\!27\)\( T + p^{31} T^{2} ) \)
41$C_2$ \( ( 1 + p^{31} T^{2} )^{2} \)
43$C_2$ \( ( 1 + \)\(17\!\cdots\!65\)\( T + p^{31} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{31} T^{2} + p^{62} T^{4} \)
53$C_2^2$ \( 1 + p^{31} T^{2} + p^{62} T^{4} \)
59$C_2^2$ \( 1 - p^{31} T^{2} + p^{62} T^{4} \)
61$C_2$ \( ( 1 - \)\(86\!\cdots\!99\)\( T + p^{31} T^{2} )( 1 - \)\(75\!\cdots\!13\)\( T + p^{31} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(78\!\cdots\!85\)\( T + p^{31} T^{2} )( 1 + \)\(38\!\cdots\!12\)\( T + p^{31} T^{2} ) \)
71$C_2$ \( ( 1 - p^{31} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(15\!\cdots\!19\)\( T + p^{31} T^{2} )( 1 - \)\(76\!\cdots\!70\)\( T + p^{31} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(35\!\cdots\!04\)\( T + p^{31} T^{2} )( 1 + \)\(50\!\cdots\!67\)\( T + p^{31} T^{2} ) \)
83$C_2$ \( ( 1 + p^{31} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{31} T^{2} + p^{62} T^{4} \)
97$C_2$ \( ( 1 - \)\(73\!\cdots\!98\)\( T + p^{31} T^{2} )( 1 + \)\(73\!\cdots\!98\)\( T + p^{31} T^{2} ) \)
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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

−12.62353885978912328473551737717, −11.54808025042274190129700650190, −10.89480972929384660825402227617, −9.987914325070084230725721168640, −9.857618411194441065726705714071, −8.709538621993976808827741191590, −8.687569246953655419150345543546, −8.277833599662984151333247277969, −7.69294964111066011318568505262, −6.66538639942885067327519578196, −6.39666040115453478095745301280, −4.95784978886825583717247975933, −4.75573230145445970656397846709, −4.05193767436502338981831546343, −3.75141612517288455401588092197, −2.72871058213817004823156770962, −2.37603376448651128012558427420, −1.81205679494473463239036725342, −1.05335691895801004512252083791, −0.42522760492814793858665364861, 0.42522760492814793858665364861, 1.05335691895801004512252083791, 1.81205679494473463239036725342, 2.37603376448651128012558427420, 2.72871058213817004823156770962, 3.75141612517288455401588092197, 4.05193767436502338981831546343, 4.75573230145445970656397846709, 4.95784978886825583717247975933, 6.39666040115453478095745301280, 6.66538639942885067327519578196, 7.69294964111066011318568505262, 8.277833599662984151333247277969, 8.687569246953655419150345543546, 8.709538621993976808827741191590, 9.857618411194441065726705714071, 9.987914325070084230725721168640, 10.89480972929384660825402227617, 11.54808025042274190129700650190, 12.62353885978912328473551737717

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