Properties

Label 40-11e60-1.1-c0e20-0-0
Degree $40$
Conductor $3.045\times 10^{62}$
Sign $1$
Analytic cond. $0.000279707$
Root an. cond. $0.815018$
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  + 3-s − 5·4-s + 5-s + 9-s − 5·12-s + 15-s + 10·16-s − 5·20-s − 4·23-s + 25-s + 27-s + 31-s − 5·36-s + 37-s + 45-s + 47-s + 10·48-s − 5·49-s + 53-s + 59-s − 5·60-s − 10·64-s − 4·67-s − 4·69-s + 71-s + 75-s + 10·80-s + ⋯
L(s)  = 1  + 3-s − 5·4-s + 5-s + 9-s − 5·12-s + 15-s + 10·16-s − 5·20-s − 4·23-s + 25-s + 27-s + 31-s − 5·36-s + 37-s + 45-s + 47-s + 10·48-s − 5·49-s + 53-s + 59-s − 5·60-s − 10·64-s − 4·67-s − 4·69-s + 71-s + 75-s + 10·80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(11^{60}\)
Sign: $1$
Analytic conductor: \(0.000279707\)
Root analytic conductor: \(0.815018\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1331} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 11^{60} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01410600819\)
\(L(\frac12)\) \(\approx\) \(0.01410600819\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
3 \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \)
5 \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \)
7 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
19 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
23 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \)
29 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
31 \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \)
37 \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \)
41 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
43 \( ( 1 - T )^{20}( 1 + T )^{20} \)
47 \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \)
53 \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \)
59 \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \)
61 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
67 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \)
71 \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
79 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
83 \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{5}( 1 + T + T^{2} + T^{3} + T^{4} )^{5} \)
89 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{4} \)
97 \( 1 - T + T^{5} - T^{6} + T^{10} - T^{12} + T^{15} - T^{17} + T^{20} - T^{23} + T^{25} - T^{28} + T^{30} - T^{34} + T^{35} - T^{39} + T^{40} \)
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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

−2.40598913735417001545858206503, −2.23662004123553542176174588292, −2.23650065638581581160866804457, −2.07114346706589476641891385131, −2.06548028031878964011245206911, −2.06059751667065391828083876081, −1.98029049798854586750929716549, −1.86561170943081062170291618874, −1.84546903389970258791212010733, −1.83874668153421823065654875332, −1.75390355149563486657709909846, −1.64217527698553263815979520721, −1.58511757954490734084886290531, −1.44555140690865457603447390606, −1.40013686751848431108800247719, −1.39127232227049611450809539498, −1.33702279595459545319028097007, −1.31359029467698936436623054699, −0.981635519143627430834342897639, −0.951139192621951683066811718955, −0.872889831212535254413303641556, −0.812435283290659250667624295587, −0.67298068428479325757427945693, −0.54819995392615220150138307518, −0.094203593162691583514429885827, 0.094203593162691583514429885827, 0.54819995392615220150138307518, 0.67298068428479325757427945693, 0.812435283290659250667624295587, 0.872889831212535254413303641556, 0.951139192621951683066811718955, 0.981635519143627430834342897639, 1.31359029467698936436623054699, 1.33702279595459545319028097007, 1.39127232227049611450809539498, 1.40013686751848431108800247719, 1.44555140690865457603447390606, 1.58511757954490734084886290531, 1.64217527698553263815979520721, 1.75390355149563486657709909846, 1.83874668153421823065654875332, 1.84546903389970258791212010733, 1.86561170943081062170291618874, 1.98029049798854586750929716549, 2.06059751667065391828083876081, 2.06548028031878964011245206911, 2.07114346706589476641891385131, 2.23650065638581581160866804457, 2.23662004123553542176174588292, 2.40598913735417001545858206503

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

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