Properties

Label 4-143e2-1.1-c0e2-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.1718876263\)
\(L(\frac12)\) \(\approx\) \(0.1718876263\)
\(L(1)\) 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

−13.58960547893136761773464901160, −12.93016617607326862101493801299, −12.59167174571363529386519262535, −11.78795778641297585343072035099, −11.68668611219368486407903566608, −10.98535866823063653794737106348, −10.62023232874391771421269454926, −10.01498252321731105275878949683, −9.246236909771999955566743720444, −9.173437165438516524913779804257, −8.425902717004282755503329240086, −8.299653780349074161218825853600, −6.88356534604153249087277566380, −6.60936187943298278927509634323, −6.17153685945518048212164616467, −5.73504655678137618143092519957, −4.55082127921418290124319818993, −3.89660235366871617514810629621, −3.16035529886055676861741520657, −1.36290338861363308272370371699, 1.36290338861363308272370371699, 3.16035529886055676861741520657, 3.89660235366871617514810629621, 4.55082127921418290124319818993, 5.73504655678137618143092519957, 6.17153685945518048212164616467, 6.60936187943298278927509634323, 6.88356534604153249087277566380, 8.299653780349074161218825853600, 8.425902717004282755503329240086, 9.173437165438516524913779804257, 9.246236909771999955566743720444, 10.01498252321731105275878949683, 10.62023232874391771421269454926, 10.98535866823063653794737106348, 11.68668611219368486407903566608, 11.78795778641297585343072035099, 12.59167174571363529386519262535, 12.93016617607326862101493801299, 13.58960547893136761773464901160

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