Properties

Label 4-21e2-1.1-c21e2-0-0
Degree $4$
Conductor $441$
Sign $1$
Analytic cond. $3444.54$
Root an. cond. $7.66095$
Motivic weight $21$
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  + 4.19e6·4-s − 1.12e9·7-s − 1.04e10·9-s + 1.31e13·16-s − 9.53e14·25-s − 4.71e15·28-s − 4.38e16·36-s + 1.15e17·37-s − 5.30e17·43-s + 7.04e17·49-s + 1.17e19·63-s + 3.68e19·64-s + 1.38e19·67-s + 3.36e20·79-s + 1.09e20·81-s − 4.00e21·100-s − 8.92e21·109-s − 1.48e22·112-s + 1.48e22·121-s + 127-s + 131-s + 137-s + 139-s − 1.38e23·144-s + 4.84e23·148-s + 149-s + 151-s + ⋯
L(s)  = 1  + 2·4-s − 1.50·7-s − 9-s + 3·16-s − 2·25-s − 3.00·28-s − 2·36-s + 3.95·37-s − 3.74·43-s + 1.26·49-s + 1.50·63-s + 4·64-s + 0.930·67-s + 3.99·79-s + 81-s − 4·100-s − 3.61·109-s − 4.51·112-s + 2·121-s − 3·144-s + 7.90·148-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(3444.54\)
Root analytic conductor: \(7.66095\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 441,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(2.520797382\)
\(L(\frac12)\) \(\approx\) \(2.520797382\)
\(L(\frac{23}{2})\) 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$C_2$ \( 1 + p^{21} T^{2} \)
7$C_2$ \( 1 + 1123983020 T + p^{21} T^{2} \)
good2$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
5$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
11$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 370076825230 T + p^{21} T^{2} )( 1 + 370076825230 T + p^{21} T^{2} ) \)
17$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 35540635313176 T + p^{21} T^{2} )( 1 + 35540635313176 T + p^{21} T^{2} ) \)
23$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
29$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 9040072142371732 T + p^{21} T^{2} )( 1 + 9040072142371732 T + p^{21} T^{2} ) \)
37$C_2$ \( ( 1 - 57776323439003290 T + p^{21} T^{2} )^{2} \)
41$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 265258444413820520 T + p^{21} T^{2} )^{2} \)
47$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
53$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
59$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10860691764464843938 T + p^{21} T^{2} )( 1 + 10860691764464843938 T + p^{21} T^{2} ) \)
67$C_2$ \( ( 1 - 6944332370266921520 T + p^{21} T^{2} )^{2} \)
71$C_2$ \( ( 1 - p^{21} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 39098623343480501290 T + p^{21} T^{2} )( 1 + 39098623343480501290 T + p^{21} T^{2} ) \)
79$C_2$ \( ( 1 - \)\(16\!\cdots\!96\)\( T + p^{21} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{21} T^{2} )^{2} \)
97$C_2$ \( ( 1 - \)\(11\!\cdots\!30\)\( T + p^{21} T^{2} )( 1 + \)\(11\!\cdots\!30\)\( T + p^{21} T^{2} ) \)
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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

−13.49569354382904190930906419692, −13.09818957488653745666804662523, −12.19200834271026321305495542484, −11.72013349629435612846346563264, −11.31691848112805478908525738983, −10.60840162707304358748483100720, −9.753470761907467476116248018365, −9.548860987763123891895986677996, −8.028631361168046090140950915120, −7.928099016901005355392369570320, −6.83550984800372671030370508226, −6.30491965200978824264409596211, −6.07978921998074230045655644739, −5.24498372983927034602958941826, −3.74850258106285630385815220731, −3.32841598919870425379346546950, −2.54615738994233115887787312201, −2.24979295557101562801861921486, −1.25900613417304301123181429083, −0.38331335069110227525029462216, 0.38331335069110227525029462216, 1.25900613417304301123181429083, 2.24979295557101562801861921486, 2.54615738994233115887787312201, 3.32841598919870425379346546950, 3.74850258106285630385815220731, 5.24498372983927034602958941826, 6.07978921998074230045655644739, 6.30491965200978824264409596211, 6.83550984800372671030370508226, 7.928099016901005355392369570320, 8.028631361168046090140950915120, 9.548860987763123891895986677996, 9.753470761907467476116248018365, 10.60840162707304358748483100720, 11.31691848112805478908525738983, 11.72013349629435612846346563264, 12.19200834271026321305495542484, 13.09818957488653745666804662523, 13.49569354382904190930906419692

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