↑

Properties

Label 4-21e2-1.1-c29e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $12518.0$
Root an. cond. $10.5775$
Motivic weight $29$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.43e7·3-s − 5.36e8·4-s − 2.49e12·7-s + 1.37e14·9-s − 7.70e15·12-s − 3.92e18·19-s − 3.58e19·21-s + 1.86e20·25-s + 9.84e20·27-s + 1.34e21·28-s − 9.48e21·31-s − 7.36e22·36-s − 7.51e22·37-s + 1.82e24·43-s + 3.01e24·49-s − 5.63e25·57-s − 1.84e26·61-s − 3.42e26·63-s + 1.54e26·64-s − 3.51e26·67-s + 3.42e27·73-s + 2.67e27·75-s + 2.10e27·76-s − 6.39e27·79-s + 4.71e27·81-s + 1.92e28·84-s − 1.36e29·93-s + ⋯
L(s)  = 1  + 1.73·3-s − 4-s − 1.39·7-s + 2·9-s − 1.73·12-s − 1.12·19-s − 2.41·21-s + 25-s + 1.73·27-s + 1.39·28-s − 2.24·31-s − 2·36-s − 1.37·37-s + 3.76·43-s + 0.936·49-s − 1.95·57-s − 2.38·61-s − 2.78·63-s + 64-s − 1.16·67-s + 3.28·73-s + 1.73·75-s + 1.12·76-s − 1.94·79-s + 81-s + 2.41·84-s − 3.89·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+29/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(12518.0\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :29/2, 29/2),\ 1)\)

Particular Values

\(L(15)\) \(\approx\) \(1.557665411\)
\(L(\frac12)\) \(\approx\) \(1.557665411\)
\(L(\frac{31}{2})\) 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$C_2$ \( 1 - p^{15} T + p^{29} T^{2} \)
7$C_2$ \( 1 + 2497125299549 T + p^{29} T^{2} \)
good2$C_2^2$ \( 1 + p^{29} T^{2} + p^{58} T^{4} \)
5$C_2^2$ \( 1 - p^{29} T^{2} + p^{58} T^{4} \)
11$C_2^2$ \( 1 + p^{29} T^{2} + p^{58} T^{4} \)
13$C_2$ \( ( 1 - 20752113277678027 T + p^{29} T^{2} )( 1 + 20752113277678027 T + p^{29} T^{2} ) \)
17$C_2^2$ \( 1 - p^{29} T^{2} + p^{58} T^{4} \)
19$C_2$ \( ( 1 - 1331063057827796129 T + p^{29} T^{2} )( 1 + 5255673528828339368 T + p^{29} T^{2} ) \)
23$C_2^2$ \( 1 + p^{29} T^{2} + p^{58} T^{4} \)
29$C_2$ \( ( 1 - p^{29} T^{2} )^{2} \)
31$C_2$ \( ( 1 + \)\(15\!\cdots\!19\)\( T + p^{29} T^{2} )( 1 + \)\(79\!\cdots\!16\)\( T + p^{29} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(31\!\cdots\!10\)\( T + p^{29} T^{2} )( 1 + \)\(10\!\cdots\!71\)\( T + p^{29} T^{2} ) \)
41$C_2$ \( ( 1 + p^{29} T^{2} )^{2} \)
43$C_2$ \( ( 1 - \)\(91\!\cdots\!45\)\( T + p^{29} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{29} T^{2} + p^{58} T^{4} \)
53$C_2^2$ \( 1 + p^{29} T^{2} + p^{58} T^{4} \)
59$C_2^2$ \( 1 - p^{29} T^{2} + p^{58} T^{4} \)
61$C_2$ \( ( 1 + \)\(36\!\cdots\!67\)\( T + p^{29} T^{2} )( 1 + \)\(14\!\cdots\!41\)\( T + p^{29} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(24\!\cdots\!55\)\( T + p^{29} T^{2} )( 1 + \)\(59\!\cdots\!36\)\( T + p^{29} T^{2} ) \)
71$C_2$ \( ( 1 - p^{29} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(20\!\cdots\!03\)\( T + p^{29} T^{2} )( 1 - \)\(13\!\cdots\!90\)\( T + p^{29} T^{2} ) \)
79$C_2$ \( ( 1 + \)\(19\!\cdots\!24\)\( T + p^{29} T^{2} )( 1 + \)\(44\!\cdots\!73\)\( T + p^{29} T^{2} ) \)
83$C_2$ \( ( 1 + p^{29} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{29} T^{2} + p^{58} T^{4} \)
97$C_2$ \( ( 1 - \)\(10\!\cdots\!66\)\( T + p^{29} T^{2} )( 1 + \)\(10\!\cdots\!66\)\( T + p^{29} T^{2} ) \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.63222986843902457616171516759, −12.44338508248290124982005653343, −10.85624882493359675891705533243, −10.55676556927689684389961847797, −9.558920341612446134062107517082, −9.318498562466087273289219318692, −8.865638993445539886912540829077, −8.492995520947261655588718357324, −7.45775076945606001523149816283, −7.20341929594517007233573798725, −6.32325682250160204156855606704, −5.61323995524880766182069813097, −4.57670981442294377327288041242, −4.18581397248839298245589176358, −3.48445742731489478000620941931, −3.18983948092927640249387278785, −2.35801538348285673396431960736, −1.94931114295699152049476924170, −0.979921407308752281584863247925, −0.26215381223237131854286688775, 0.26215381223237131854286688775, 0.979921407308752281584863247925, 1.94931114295699152049476924170, 2.35801538348285673396431960736, 3.18983948092927640249387278785, 3.48445742731489478000620941931, 4.18581397248839298245589176358, 4.57670981442294377327288041242, 5.61323995524880766182069813097, 6.32325682250160204156855606704, 7.20341929594517007233573798725, 7.45775076945606001523149816283, 8.492995520947261655588718357324, 8.865638993445539886912540829077, 9.318498562466087273289219318692, 9.558920341612446134062107517082, 10.55676556927689684389961847797, 10.85624882493359675891705533243, 12.44338508248290124982005653343, 12.63222986843902457616171516759

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