Properties

Label 4-21e2-1.1-c32e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $18555.8$
Root an. cond. $11.6713$
Motivic weight $32$
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 − 4.29e9·4-s + 6.51e13·7-s + 1.84e17·12-s + 2.44e17·13-s + 7.82e18·19-s − 2.80e21·21-s − 2.32e22·25-s + 7.97e22·27-s − 2.79e23·28-s + 1.41e24·31-s + 2.15e25·37-s − 1.05e25·39-s + 5.45e26·43-s + 3.14e27·49-s − 1.04e27·52-s − 3.36e26·57-s + 3.95e28·61-s + 7.92e28·64-s − 5.31e27·67-s − 7.67e29·73-s + 1.00e30·75-s − 3.35e28·76-s − 2.02e30·79-s − 3.43e30·81-s + 1.20e31·84-s + 1.59e31·91-s + ⋯
L(s)  = 1  − 3-s − 4-s + 1.96·7-s + 12-s + 0.367·13-s + 0.0271·19-s − 1.96·21-s − 25-s + 27-s − 1.96·28-s + 1.94·31-s + 1.74·37-s − 0.367·39-s + 3.99·43-s + 2.84·49-s − 0.367·52-s − 0.0271·57-s + 1.07·61-s + 64-s − 0.0322·67-s − 1.18·73-s + 75-s − 0.0271·76-s − 0.880·79-s − 81-s + 1.96·84-s + 0.719·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+16)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(18555.8\)
Root analytic conductor: \(11.6713\)
Motivic weight: \(32\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :16, 16),\ 1)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(2.628842828\)
\(L(\frac12)\) \(\approx\) \(2.628842828\)
\(L(17)\) 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$C_2$ \( 1 + p^{16} T + p^{32} T^{2} \)
7$C_2$ \( 1 - 65151959156159 T + p^{32} T^{2} \)
good2$C_2$ \( ( 1 - p^{16} T + p^{32} T^{2} )( 1 + p^{16} T + p^{32} T^{2} ) \)
5$C_2$ \( ( 1 - p^{16} T + p^{32} T^{2} )( 1 + p^{16} T + p^{32} T^{2} ) \)
11$C_2$ \( ( 1 - p^{16} T + p^{32} T^{2} )( 1 + p^{16} T + p^{32} T^{2} ) \)
13$C_2$ \( ( 1 - 122181402203975039 T + p^{32} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{16} T + p^{32} T^{2} )( 1 + p^{16} T + p^{32} T^{2} ) \)
19$C_2$ \( ( 1 - \)\(50\!\cdots\!74\)\( T + p^{32} T^{2} )( 1 + \)\(49\!\cdots\!13\)\( T + p^{32} T^{2} ) \)
23$C_2$ \( ( 1 - p^{16} T + p^{32} T^{2} )( 1 + p^{16} T + p^{32} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{16} T )^{2}( 1 + p^{16} T )^{2} \)
31$C_2$ \( ( 1 - \)\(10\!\cdots\!54\)\( T + p^{32} T^{2} )( 1 - \)\(40\!\cdots\!59\)\( T + p^{32} T^{2} ) \)
37$C_2$ \( ( 1 - \)\(21\!\cdots\!82\)\( T + p^{32} T^{2} )( 1 - \)\(37\!\cdots\!79\)\( T + p^{32} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{16} T )^{2}( 1 + p^{16} T )^{2} \)
43$C_2$ \( ( 1 - \)\(27\!\cdots\!27\)\( T + p^{32} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{16} T + p^{32} T^{2} )( 1 + p^{16} T + p^{32} T^{2} ) \)
53$C_2$ \( ( 1 - p^{16} T + p^{32} T^{2} )( 1 + p^{16} T + p^{32} T^{2} ) \)
59$C_2$ \( ( 1 - p^{16} T + p^{32} T^{2} )( 1 + p^{16} T + p^{32} T^{2} ) \)
61$C_2$ \( ( 1 - \)\(73\!\cdots\!47\)\( T + p^{32} T^{2} )( 1 + \)\(33\!\cdots\!21\)\( T + p^{32} T^{2} ) \)
67$C_2$ \( ( 1 - \)\(28\!\cdots\!54\)\( T + p^{32} T^{2} )( 1 + \)\(28\!\cdots\!13\)\( T + p^{32} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{16} T )^{2}( 1 + p^{16} T )^{2} \)
73$C_2$ \( ( 1 - \)\(52\!\cdots\!22\)\( T + p^{32} T^{2} )( 1 + \)\(12\!\cdots\!01\)\( T + p^{32} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(25\!\cdots\!42\)\( T + p^{32} T^{2} )( 1 + \)\(45\!\cdots\!81\)\( T + p^{32} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{16} T )^{2}( 1 + p^{16} T )^{2} \)
89$C_2$ \( ( 1 - p^{16} T + p^{32} T^{2} )( 1 + p^{16} T + p^{32} T^{2} ) \)
97$C_2$ \( ( 1 - \)\(12\!\cdots\!14\)\( T + p^{32} T^{2} )^{2} \)
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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

−11.93123616594753840638631511863, −11.25977208872065284165053789426, −11.22404101111873193513511153439, −10.38839585772668743983918857480, −9.761494789861261556376771884386, −8.795514513826156545547640106280, −8.689233516055067288426319978044, −7.67351451001015672166724938868, −7.63279756437076862540639556802, −6.33967424979526184532554768217, −5.88955765549239595922325814973, −5.34329495517252849127077522964, −4.70701965937614222843312961314, −4.34519796733871313121492236275, −3.95482219675028789688841558724, −2.63899092748524125086599709242, −2.27803599298492874428520824623, −1.25463829816007482329923375189, −0.879249101903780611763628975886, −0.48501811786860854016447992974, 0.48501811786860854016447992974, 0.879249101903780611763628975886, 1.25463829816007482329923375189, 2.27803599298492874428520824623, 2.63899092748524125086599709242, 3.95482219675028789688841558724, 4.34519796733871313121492236275, 4.70701965937614222843312961314, 5.34329495517252849127077522964, 5.88955765549239595922325814973, 6.33967424979526184532554768217, 7.63279756437076862540639556802, 7.67351451001015672166724938868, 8.689233516055067288426319978044, 8.795514513826156545547640106280, 9.761494789861261556376771884386, 10.38839585772668743983918857480, 11.22404101111873193513511153439, 11.25977208872065284165053789426, 11.93123616594753840638631511863

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