Properties

Label 4-21e2-1.1-c22e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $4148.46$
Root an. cond. $8.02549$
Motivic weight $22$
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  + 1.77e5·3-s − 4.19e6·4-s + 1.99e9·7-s − 7.43e11·12-s + 4.18e12·13-s − 2.06e13·19-s + 3.53e14·21-s − 2.38e15·25-s − 5.55e15·27-s − 8.37e15·28-s − 3.70e16·31-s − 8.44e16·37-s + 7.42e17·39-s + 2.53e18·43-s + 8.19e16·49-s − 1.75e19·52-s − 3.66e18·57-s + 6.73e19·61-s + 7.37e19·64-s − 2.14e20·67-s − 6.21e20·73-s − 4.22e20·75-s + 8.67e19·76-s + 1.03e21·79-s − 9.84e20·81-s − 1.48e21·84-s + 8.36e21·91-s + ⋯
L(s)  = 1  + 3-s − 4-s + 1.01·7-s − 12-s + 2.33·13-s − 0.177·19-s + 1.01·21-s − 25-s − 27-s − 1.01·28-s − 1.45·31-s − 0.474·37-s + 2.33·39-s + 2.73·43-s + 0.0209·49-s − 2.33·52-s − 0.177·57-s + 1.54·61-s + 64-s − 1.75·67-s − 1.98·73-s − 75-s + 0.177·76-s + 1.38·79-s − 81-s − 1.01·84-s + 2.36·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+11)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4148.46\)
Root analytic conductor: \(8.02549\)
Motivic weight: \(22\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :11, 11),\ 1)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(3.046763515\)
\(L(\frac12)\) \(\approx\) \(3.046763515\)
\(L(12)\) 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$C_2$ \( 1 - p^{11} T + p^{22} T^{2} \)
7$C_2$ \( 1 - 1997931923 T + p^{22} T^{2} \)
good2$C_2$ \( ( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} ) \)
5$C_2$ \( ( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} ) \)
11$C_2$ \( ( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} ) \)
13$C_2$ \( ( 1 - 2094315786047 T + p^{22} T^{2} )^{2} \)
17$C_2$ \( ( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} ) \)
19$C_2$ \( ( 1 - 190634130977363 T + p^{22} T^{2} )( 1 + 211308066581014 T + p^{22} T^{2} ) \)
23$C_2$ \( ( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{11} T )^{2}( 1 + p^{11} T )^{2} \)
31$C_2$ \( ( 1 - 11550498557326034 T + p^{22} T^{2} )( 1 + 48632122495141621 T + p^{22} T^{2} ) \)
37$C_2$ \( ( 1 - 257128060064352074 T + p^{22} T^{2} )( 1 + 341584946762516377 T + p^{22} T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{11} T )^{2}( 1 + p^{11} T )^{2} \)
43$C_2$ \( ( 1 - 1268503014678232811 T + p^{22} T^{2} )^{2} \)
47$C_2$ \( ( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} ) \)
53$C_2$ \( ( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} ) \)
59$C_2$ \( ( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} ) \)
61$C_2$ \( ( 1 - 81382296126928348247 T + p^{22} T^{2} )( 1 + 13987693501999930801 T + p^{22} T^{2} ) \)
67$C_2$ \( ( 1 + 5287275662394476662 T + p^{22} T^{2} )( 1 + \)\(20\!\cdots\!41\)\( T + p^{22} T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{11} T )^{2}( 1 + p^{11} T )^{2} \)
73$C_2$ \( ( 1 + \)\(23\!\cdots\!54\)\( T + p^{22} T^{2} )( 1 + \)\(38\!\cdots\!53\)\( T + p^{22} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(14\!\cdots\!11\)\( T + p^{22} T^{2} )( 1 + \)\(41\!\cdots\!42\)\( T + p^{22} T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - p^{11} T )^{2}( 1 + p^{11} T )^{2} \)
89$C_2$ \( ( 1 - p^{11} T + p^{22} T^{2} )( 1 + p^{11} T + p^{22} T^{2} ) \)
97$C_2$ \( ( 1 + \)\(16\!\cdots\!42\)\( T + p^{22} T^{2} )^{2} \)
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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

−13.43683224837033743629287262401, −13.37517800005923698767915862728, −12.44653999021696057947110473897, −11.36621168039621992412375261579, −11.12816489753663890751407393301, −10.32887734908971639787694117015, −9.172345541004213088969310451312, −9.065689293554382311384110203527, −8.411835390081561262960752756214, −7.971420913561324854088479446579, −7.22247068488377560410173538599, −5.99321389419499714149232001011, −5.62837798200384204957780300273, −4.63385109819513583727593550331, −3.82347155259651740531913589461, −3.73641341379256289259871733339, −2.62096946047486632472174030516, −1.81991798293470538725445347445, −1.27399615008367403180032279431, −0.42523232211675483034127245658, 0.42523232211675483034127245658, 1.27399615008367403180032279431, 1.81991798293470538725445347445, 2.62096946047486632472174030516, 3.73641341379256289259871733339, 3.82347155259651740531913589461, 4.63385109819513583727593550331, 5.62837798200384204957780300273, 5.99321389419499714149232001011, 7.22247068488377560410173538599, 7.971420913561324854088479446579, 8.411835390081561262960752756214, 9.065689293554382311384110203527, 9.172345541004213088969310451312, 10.32887734908971639787694117015, 11.12816489753663890751407393301, 11.36621168039621992412375261579, 12.44653999021696057947110473897, 13.37517800005923698767915862728, 13.43683224837033743629287262401

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