Properties

Label 40-1503e20-1.1-c0e20-0-0
Degree $40$
Conductor $3.461\times 10^{63}$
Sign $1$
Analytic cond. $0.00317923$
Root an. cond. $0.866080$
Motivic weight $0$
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  − 2-s + 3-s − 6-s − 7-s − 8-s + 9-s − 11-s + 14-s + 16-s − 18-s + 2·19-s − 21-s + 22-s − 24-s − 10·25-s − 29-s − 31-s − 33-s − 2·38-s + 42-s − 47-s + 48-s + 10·50-s + 56-s + 2·57-s + 58-s + 2·61-s + ⋯
L(s)  = 1  − 2-s + 3-s − 6-s − 7-s − 8-s + 9-s − 11-s + 14-s + 16-s − 18-s + 2·19-s − 21-s + 22-s − 24-s − 10·25-s − 29-s − 31-s − 33-s − 2·38-s + 42-s − 47-s + 48-s + 10·50-s + 56-s + 2·57-s + 58-s + 2·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 167^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 167^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(3^{40} \cdot 167^{20}\)
Sign: $1$
Analytic conductor: \(0.00317923\)
Root analytic conductor: \(0.866080\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1503} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 3^{40} \cdot 167^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03797477876\)
\(L(\frac12)\) \(\approx\) \(0.03797477876\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} \)
167 \( ( 1 + T + T^{2} )^{10} \)
good2 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
5 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
7 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
11 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
13 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
17 \( ( 1 - T )^{20}( 1 + T )^{20} \)
19 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
23 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
31 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
37 \( ( 1 - T )^{20}( 1 + T )^{20} \)
41 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
43 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
47 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} ) \)
53 \( ( 1 - T )^{20}( 1 + T )^{20} \)
59 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
61 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
67 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
71 \( ( 1 - T )^{20}( 1 + T )^{20} \)
73 \( ( 1 - T )^{20}( 1 + T )^{20} \)
79 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
83 \( ( 1 - T + T^{2} )^{10}( 1 + T + T^{2} )^{10} \)
89 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
97 \( ( 1 - T + T^{3} - T^{4} + T^{6} - T^{7} + T^{9} - T^{10} + T^{11} - T^{13} + T^{14} - T^{16} + T^{17} - T^{19} + T^{20} )^{2} \)
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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

−2.17530743654655514447522587235, −2.14396614264764320861934669105, −2.12411195596493788798965879338, −2.12209605220003440653709383077, −2.11404663261894957064802076386, −2.10374374832451426694324189210, −2.09485811408538418310023457044, −2.04941161487163597099571154830, −1.78216389281521211174669203058, −1.70323546421535368409084072573, −1.60730189065608048597452288240, −1.59288647317470438617910280532, −1.56395743915187067037306007022, −1.54964037857960168108665446835, −1.45925005899777842929935703909, −1.40346178543609988657623059464, −1.37880262668379416661136073264, −1.20605069411994647896840475068, −1.01877374194905198550975234450, −0.878050014798597375235898463768, −0.825473545068127294897526248215, −0.78719637920725477006538219327, −0.65698921081762489730133640662, −0.39511257561634008756486624954, −0.15495342712190889598668668894, 0.15495342712190889598668668894, 0.39511257561634008756486624954, 0.65698921081762489730133640662, 0.78719637920725477006538219327, 0.825473545068127294897526248215, 0.878050014798597375235898463768, 1.01877374194905198550975234450, 1.20605069411994647896840475068, 1.37880262668379416661136073264, 1.40346178543609988657623059464, 1.45925005899777842929935703909, 1.54964037857960168108665446835, 1.56395743915187067037306007022, 1.59288647317470438617910280532, 1.60730189065608048597452288240, 1.70323546421535368409084072573, 1.78216389281521211174669203058, 2.04941161487163597099571154830, 2.09485811408538418310023457044, 2.10374374832451426694324189210, 2.11404663261894957064802076386, 2.12209605220003440653709383077, 2.12411195596493788798965879338, 2.14396614264764320861934669105, 2.17530743654655514447522587235

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

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