Properties

Label 4-143e2-1.1-c0e2-0-1
Degree $4$
Conductor $20449$
Sign $1$
Analytic cond. $0.00509314$
Root an. cond. $0.267144$
Motivic weight $0$
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  + 2-s − 3-s − 6-s + 7-s − 2·11-s − 2·13-s + 14-s + 19-s − 21-s − 2·22-s − 23-s + 2·25-s − 2·26-s − 32-s + 2·33-s + 38-s + 2·39-s + 41-s − 42-s − 46-s + 2·50-s − 53-s − 57-s − 64-s + 2·66-s + 69-s + 73-s + ⋯
L(s)  = 1  + 2-s − 3-s − 6-s + 7-s − 2·11-s − 2·13-s + 14-s + 19-s − 21-s − 2·22-s − 23-s + 2·25-s − 2·26-s − 32-s + 2·33-s + 38-s + 2·39-s + 41-s − 42-s − 46-s + 2·50-s − 53-s − 57-s − 64-s + 2·66-s + 69-s + 73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20449\)    =    \(11^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.00509314\)
Root analytic conductor: \(0.267144\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{143} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 20449,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4370540869\)
\(L(\frac12)\) \(\approx\) \(0.4370540869\)
\(L(1)\) 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)$
bad11$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
3$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
5$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
7$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
23$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
53$C_4$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
83$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−13.60486632803815923946613303662, −13.01428957171829116532910775374, −12.49653956081239081415209781238, −12.36739753161912299402971421224, −11.67030199132299516852202375700, −11.09964703010420586370456732131, −10.82066324397992018341842346619, −10.21318557697500063287799539775, −9.724324737236978039854594585250, −9.004257973228336367496572581079, −8.129979544218070959974762466048, −7.66026602909536586139046396552, −7.34040097683464702793451134111, −6.44399786173049336138211999786, −5.40644236408523558557252902644, −5.31421961759558757154414127899, −4.86221255192137125487916254236, −4.38802451253424855324823205856, −3.07141793703332287184327843999, −2.27203371878102644767358401045, 2.27203371878102644767358401045, 3.07141793703332287184327843999, 4.38802451253424855324823205856, 4.86221255192137125487916254236, 5.31421961759558757154414127899, 5.40644236408523558557252902644, 6.44399786173049336138211999786, 7.34040097683464702793451134111, 7.66026602909536586139046396552, 8.129979544218070959974762466048, 9.004257973228336367496572581079, 9.724324737236978039854594585250, 10.21318557697500063287799539775, 10.82066324397992018341842346619, 11.09964703010420586370456732131, 11.67030199132299516852202375700, 12.36739753161912299402971421224, 12.49653956081239081415209781238, 13.01428957171829116532910775374, 13.60486632803815923946613303662

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