Properties

Label 40-1412e20-1.1-c0e20-0-0
Degree $40$
Conductor $9.924\times 10^{62}$
Sign $1$
Analytic cond. $0.000911667$
Root an. cond. $0.839452$
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  + 2·2-s + 4-s − 2·5-s − 4·10-s − 2·13-s + 4·17-s − 2·20-s + 2·25-s − 4·26-s + 8·34-s − 20·37-s + 4·50-s − 2·52-s − 2·53-s + 4·65-s + 4·68-s − 40·74-s + 81-s − 8·85-s − 2·89-s − 4·97-s + 2·100-s − 2·101-s − 4·106-s + 18·109-s − 2·121-s − 2·125-s + ⋯
L(s)  = 1  + 2·2-s + 4-s − 2·5-s − 4·10-s − 2·13-s + 4·17-s − 2·20-s + 2·25-s − 4·26-s + 8·34-s − 20·37-s + 4·50-s − 2·52-s − 2·53-s + 4·65-s + 4·68-s − 40·74-s + 81-s − 8·85-s − 2·89-s − 4·97-s + 2·100-s − 2·101-s − 4·106-s + 18·109-s − 2·121-s − 2·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 353^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 353^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{40} \cdot 353^{20}\)
Sign: $1$
Analytic conductor: \(0.000911667\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1412} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{40} \cdot 353^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2672199859\)
\(L(\frac12)\) \(\approx\) \(0.2672199859\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
353 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
good3 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
5 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
7 \( ( 1 + T^{4} )^{10} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
13 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{4} \)
19 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
23 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
29 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
31 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
37 \( ( 1 + T )^{20}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
41 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
43 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
47 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
53 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
59 \( ( 1 + T^{4} )^{10} \)
61 \( ( 1 + T^{2} )^{10}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
67 \( ( 1 + T^{4} )^{10} \)
71 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
73 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
79 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
97 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.32698867437060913901012158724, −2.22255141606517467599373799573, −2.21293579432034119608245929595, −2.19885374668561098438286554156, −2.07135298406014991774317380160, −2.05938533962624588179913753855, −1.99177430923159548210765465850, −1.89539910376946067184207942033, −1.80129263157510118824719142101, −1.67912652368452641011649315106, −1.67789766662466335833982943329, −1.62795174909827885768338659033, −1.58588897563417691520473191964, −1.48207416813000266745294134238, −1.44154676304798174163277548108, −1.38029605343241963760618271762, −1.36508547971127231589829636037, −1.33898727833752210470641386358, −1.24799396163651827284621073662, −1.11736088699219753603585621097, −1.02244064340270583080891420203, −0.61072286260833724208011865315, −0.55217582369366805732480787694, −0.45713985828832129268359005991, −0.28586819622339563061277034334, 0.28586819622339563061277034334, 0.45713985828832129268359005991, 0.55217582369366805732480787694, 0.61072286260833724208011865315, 1.02244064340270583080891420203, 1.11736088699219753603585621097, 1.24799396163651827284621073662, 1.33898727833752210470641386358, 1.36508547971127231589829636037, 1.38029605343241963760618271762, 1.44154676304798174163277548108, 1.48207416813000266745294134238, 1.58588897563417691520473191964, 1.62795174909827885768338659033, 1.67789766662466335833982943329, 1.67912652368452641011649315106, 1.80129263157510118824719142101, 1.89539910376946067184207942033, 1.99177430923159548210765465850, 2.05938533962624588179913753855, 2.07135298406014991774317380160, 2.19885374668561098438286554156, 2.21293579432034119608245929595, 2.22255141606517467599373799573, 2.32698867437060913901012158724

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

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