Properties

Degree 20
Conductor $ 2^{20} \cdot 3^{10} \cdot 167^{10} $
Sign $1$
Motivic weight 0
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 + 2·11-s − 2·22-s − 10·25-s − 2·33-s + 2·47-s − 49-s + 10·50-s + 2·61-s + 2·66-s + 10·75-s − 2·94-s − 2·97-s + 98-s + 2·107-s + 121-s − 2·122-s + 127-s + 131-s + 137-s + 139-s − 2·141-s + 147-s + 149-s − 10·150-s + ⋯
L(s)  = 1  − 2-s − 3-s + 6-s + 2·11-s − 2·22-s − 10·25-s − 2·33-s + 2·47-s − 49-s + 10·50-s + 2·61-s + 2·66-s + 10·75-s − 2·94-s − 2·97-s + 98-s + 2·107-s + 121-s − 2·122-s + 127-s + 131-s + 137-s + 139-s − 2·141-s + 147-s + 149-s − 10·150-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 167^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{10} \cdot 167^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(20\)
\( N \)  =  \(2^{20} \cdot 3^{10} \cdot 167^{10}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  induced by $\chi_{2004} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((20,\ 2^{20} \cdot 3^{10} \cdot 167^{10} ,\ ( \ : [0]^{10} ),\ 1 )\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.2463203317\)
\(L(\frac12)\)  \(\approx\)  \(0.2463203317\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\(F_p(T)\) is a polynomial of degree 20. If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 19.
$p$$F_p(T)$
bad2 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
3 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \)
167 \( ( 1 - T )^{10} \)
good5 \( ( 1 + T^{2} )^{10} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
13 \( ( 1 - T )^{10}( 1 + T )^{10} \)
17 \( ( 1 + T^{2} )^{10} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
23 \( ( 1 - T )^{10}( 1 + T )^{10} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
37 \( ( 1 - T )^{10}( 1 + T )^{10} \)
41 \( ( 1 + T^{2} )^{10} \)
43 \( ( 1 + T^{2} )^{10} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
53 \( ( 1 + T^{2} )^{10} \)
59 \( ( 1 - T )^{10}( 1 + T )^{10} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
67 \( ( 1 + T^{2} )^{10} \)
71 \( ( 1 - T )^{10}( 1 + T )^{10} \)
73 \( ( 1 - T )^{10}( 1 + T )^{10} \)
79 \( ( 1 + T^{2} )^{10} \)
83 \( ( 1 - T )^{10}( 1 + T )^{10} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} ) \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−3.43062018570299521503068427378, −3.37295572007839185449154365206, −3.33141579554695685553270990557, −3.19567922589099793609408758770, −3.14691847581597375063419529901, −3.03006468940317916439329839449, −2.91223632622808443979817457861, −2.79893431239952689024205794906, −2.42271863402911549332038774359, −2.40030282703516699244742858961, −2.39591821652881302389623929964, −2.25496321622871098715489189879, −2.15659483998477071812008552309, −2.07722669255879354796332111870, −1.77534703422381633789423465072, −1.76741432591720677145242629113, −1.74375930733563195692064760898, −1.67234156056442930718178238898, −1.58219327958329133255244849674, −1.47650705043448272081151416953, −1.19323653369796004732161807247, −0.821356546807012871468183666687, −0.71142446747013975391390430563, −0.48935273346476753254294368942, −0.44030265948757455079775548392, 0.44030265948757455079775548392, 0.48935273346476753254294368942, 0.71142446747013975391390430563, 0.821356546807012871468183666687, 1.19323653369796004732161807247, 1.47650705043448272081151416953, 1.58219327958329133255244849674, 1.67234156056442930718178238898, 1.74375930733563195692064760898, 1.76741432591720677145242629113, 1.77534703422381633789423465072, 2.07722669255879354796332111870, 2.15659483998477071812008552309, 2.25496321622871098715489189879, 2.39591821652881302389623929964, 2.40030282703516699244742858961, 2.42271863402911549332038774359, 2.79893431239952689024205794906, 2.91223632622808443979817457861, 3.03006468940317916439329839449, 3.14691847581597375063419529901, 3.19567922589099793609408758770, 3.33141579554695685553270990557, 3.37295572007839185449154365206, 3.43062018570299521503068427378

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

Plot not available for L-functions of degree greater than 10.