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Properties

Label 4-21e2-1.1-c37e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $33160.2$
Root an. cond. $13.4944$
Motivic weight $37$
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  − 1.16e9·3-s − 1.37e11·4-s + 5.88e15·7-s + 9.00e17·9-s + 1.59e20·12-s + 1.56e24·19-s − 6.83e24·21-s + 7.27e25·25-s − 5.23e26·27-s − 8.08e26·28-s + 1.34e28·31-s − 1.23e29·36-s − 1.98e28·37-s − 6.25e30·43-s + 1.60e31·49-s − 1.82e33·57-s − 3.55e33·61-s + 5.29e33·63-s + 2.59e33·64-s − 1.02e34·67-s + 1.02e35·73-s − 8.45e34·75-s − 2.15e35·76-s − 2.23e35·79-s + 2.02e35·81-s + 9.39e35·84-s − 1.56e37·93-s + ⋯
L(s)  = 1  − 1.73·3-s − 4-s + 1.36·7-s + 2·9-s + 1.73·12-s + 3.45·19-s − 2.36·21-s + 25-s − 1.73·27-s − 1.36·28-s + 3.46·31-s − 2·36-s − 0.193·37-s − 3.77·43-s + 0.864·49-s − 5.98·57-s − 3.33·61-s + 2.73·63-s + 64-s − 1.69·67-s + 3.44·73-s − 1.73·75-s − 3.45·76-s − 1.75·79-s + 81-s + 2.36·84-s − 5.99·93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+37/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(33160.2\)
Root analytic conductor: \(13.4944\)
Motivic weight: \(37\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :37/2, 37/2),\ 1)\)

Particular Values

\(L(19)\) \(\approx\) \(2.288052861\)
\(L(\frac12)\) \(\approx\) \(2.288052861\)
\(L(\frac{39}{2})\) 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$C_2$ \( 1 + p^{19} T + p^{37} T^{2} \)
7$C_2$ \( 1 - 5882725491086809 T + p^{37} T^{2} \)
good2$C_2^2$ \( 1 + p^{37} T^{2} + p^{74} T^{4} \)
5$C_2^2$ \( 1 - p^{37} T^{2} + p^{74} T^{4} \)
11$C_2^2$ \( 1 + p^{37} T^{2} + p^{74} T^{4} \)
13$C_2$ \( ( 1 - \)\(73\!\cdots\!37\)\( T + p^{37} T^{2} )( 1 + \)\(73\!\cdots\!37\)\( T + p^{37} T^{2} ) \)
17$C_2^2$ \( 1 - p^{37} T^{2} + p^{74} T^{4} \)
19$C_2$ \( ( 1 - \)\(81\!\cdots\!59\)\( T + p^{37} T^{2} )( 1 - \)\(75\!\cdots\!52\)\( T + p^{37} T^{2} ) \)
23$C_2^2$ \( 1 + p^{37} T^{2} + p^{74} T^{4} \)
29$C_2$ \( ( 1 - p^{37} T^{2} )^{2} \)
31$C_2$ \( ( 1 - \)\(68\!\cdots\!71\)\( T + p^{37} T^{2} )( 1 - \)\(66\!\cdots\!84\)\( T + p^{37} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(16\!\cdots\!61\)\( T + p^{37} T^{2} )( 1 + \)\(18\!\cdots\!70\)\( T + p^{37} T^{2} ) \)
41$C_2$ \( ( 1 + p^{37} T^{2} )^{2} \)
43$C_2$ \( ( 1 + \)\(31\!\cdots\!65\)\( T + p^{37} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{37} T^{2} + p^{74} T^{4} \)
53$C_2^2$ \( 1 + p^{37} T^{2} + p^{74} T^{4} \)
59$C_2^2$ \( 1 - p^{37} T^{2} + p^{74} T^{4} \)
61$C_2$ \( ( 1 + \)\(14\!\cdots\!01\)\( T + p^{37} T^{2} )( 1 + \)\(20\!\cdots\!47\)\( T + p^{37} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(42\!\cdots\!96\)\( T + p^{37} T^{2} )( 1 + \)\(10\!\cdots\!35\)\( T + p^{37} T^{2} ) \)
71$C_2$ \( ( 1 - p^{37} T^{2} )^{2} \)
73$C_2$ \( ( 1 - \)\(53\!\cdots\!70\)\( T + p^{37} T^{2} )( 1 - \)\(48\!\cdots\!07\)\( T + p^{37} T^{2} ) \)
79$C_2$ \( ( 1 + \)\(51\!\cdots\!23\)\( T + p^{37} T^{2} )( 1 + \)\(21\!\cdots\!44\)\( T + p^{37} T^{2} ) \)
83$C_2$ \( ( 1 + p^{37} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{37} T^{2} + p^{74} T^{4} \)
97$C_2$ \( ( 1 - \)\(94\!\cdots\!14\)\( T + p^{37} T^{2} )( 1 + \)\(94\!\cdots\!14\)\( T + p^{37} T^{2} ) \)
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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

−11.46451520800762463843753189871, −11.26956508758484783160755901112, −10.17104326271434930368895140009, −10.09172469543090865734279668061, −9.362158840709763974524425570880, −8.596372921485277732339030703105, −7.971459471117542931949314887431, −7.48611697848609999479237306604, −6.72631571756274615824458733103, −6.24528152846207939646504847702, −5.36907821426381340439569147469, −5.01918170998941696182132812785, −4.78742299386941547732331703292, −4.40294455097480153695639171462, −3.30707521816141849256237759062, −2.92213030573735521262456048016, −1.55261749067565702897214463170, −1.41239151959645660431877604177, −0.72493513179751077437943767995, −0.50151550829269452294415441616, 0.50151550829269452294415441616, 0.72493513179751077437943767995, 1.41239151959645660431877604177, 1.55261749067565702897214463170, 2.92213030573735521262456048016, 3.30707521816141849256237759062, 4.40294455097480153695639171462, 4.78742299386941547732331703292, 5.01918170998941696182132812785, 5.36907821426381340439569147469, 6.24528152846207939646504847702, 6.72631571756274615824458733103, 7.48611697848609999479237306604, 7.971459471117542931949314887431, 8.596372921485277732339030703105, 9.362158840709763974524425570880, 10.09172469543090865734279668061, 10.17104326271434930368895140009, 11.26956508758484783160755901112, 11.46451520800762463843753189871

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