Properties

Label 40-979e20-1.1-c0e20-0-0
Degree $40$
Conductor $6.541\times 10^{59}$
Sign $1$
Analytic cond. $6.00891\times 10^{-7}$
Root an. cond. $0.698988$
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  + 2·3-s − 2·4-s − 9·9-s − 2·11-s − 4·12-s + 16-s − 2·23-s + 2·25-s − 20·27-s + 2·31-s − 4·33-s + 18·36-s + 2·37-s + 4·44-s + 2·48-s − 2·59-s − 4·69-s + 4·75-s + 45·81-s + 2·89-s + 4·92-s + 4·93-s + 18·99-s − 4·100-s − 2·103-s + 40·108-s + 4·111-s + ⋯
L(s)  = 1  + 2·3-s − 2·4-s − 9·9-s − 2·11-s − 4·12-s + 16-s − 2·23-s + 2·25-s − 20·27-s + 2·31-s − 4·33-s + 18·36-s + 2·37-s + 4·44-s + 2·48-s − 2·59-s − 4·69-s + 4·75-s + 45·81-s + 2·89-s + 4·92-s + 4·93-s + 18·99-s − 4·100-s − 2·103-s + 40·108-s + 4·111-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{20} \cdot 89^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{20} \cdot 89^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(11^{20} \cdot 89^{20}\)
Sign: $1$
Analytic conductor: \(6.00891\times 10^{-7}\)
Root analytic conductor: \(0.698988\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{979} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 11^{20} \cdot 89^{20} ,\ ( \ : [0]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04612848055\)
\(L(\frac12)\) \(\approx\) \(0.04612848055\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
3 \( ( 1 + T^{2} )^{10}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2} \)
5 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
7 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
13 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
17 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
19 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
23 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
29 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
31 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
41 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
43 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
53 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
59 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
61 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
67 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
71 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
73 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2} \)
79 \( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} )^{2} \)
83 \( 1 - T^{4} + T^{8} - T^{12} + T^{16} - T^{20} + T^{24} - T^{28} + T^{32} - T^{36} + T^{40} \)
97 \( ( 1 + T^{2} )^{10}( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} ) \)
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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

−2.57858927409333867078556511237, −2.52237379167063061856921035295, −2.41064286106778154487962673482, −2.39271990228070067458945692909, −2.31356689171269121995816211894, −2.30373759594896768913227556557, −2.19026682655106731776980025373, −2.13182566736330957078099474549, −2.11391224914818275159478963833, −2.09626202649743980517501277663, −2.08705832437648798577047303009, −2.01210836380621190924064756594, −1.68532820954735934118315712875, −1.66136317668370935388461041343, −1.65668654032844756327737797351, −1.35881746677782194670610626476, −1.33381930072042537885597664116, −1.32399139404316537792266009428, −1.16056155558288583597169496895, −1.05178073060417845049947863914, −0.987660240979724868060312152079, −0.71801525204924389909794120623, −0.53971717709319661149442619472, −0.49830730358581025940857390209, −0.45068107901594621129687256700, 0.45068107901594621129687256700, 0.49830730358581025940857390209, 0.53971717709319661149442619472, 0.71801525204924389909794120623, 0.987660240979724868060312152079, 1.05178073060417845049947863914, 1.16056155558288583597169496895, 1.32399139404316537792266009428, 1.33381930072042537885597664116, 1.35881746677782194670610626476, 1.65668654032844756327737797351, 1.66136317668370935388461041343, 1.68532820954735934118315712875, 2.01210836380621190924064756594, 2.08705832437648798577047303009, 2.09626202649743980517501277663, 2.11391224914818275159478963833, 2.13182566736330957078099474549, 2.19026682655106731776980025373, 2.30373759594896768913227556557, 2.31356689171269121995816211894, 2.39271990228070067458945692909, 2.41064286106778154487962673482, 2.52237379167063061856921035295, 2.57858927409333867078556511237

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

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