Properties

Label 40-384e20-1.1-c2e20-0-1
Degree $40$
Conductor $4.860\times 10^{51}$
Sign $1$
Analytic cond. $2.47357\times 10^{20}$
Root an. cond. $3.23469$
Motivic weight $2$
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  + 6·3-s + 18·9-s − 92·13-s + 52·19-s + 30·27-s − 80·31-s + 116·37-s − 552·39-s − 172·43-s + 308·49-s + 312·57-s + 244·61-s − 356·67-s + 384·79-s − 29·81-s − 480·93-s + 472·97-s − 156·109-s + 696·111-s − 1.65e3·117-s + 127-s − 1.03e3·129-s + 131-s + 137-s + 139-s + 1.84e3·147-s + 149-s + ⋯
L(s)  = 1  + 2·3-s + 2·9-s − 7.07·13-s + 2.73·19-s + 10/9·27-s − 2.58·31-s + 3.13·37-s − 14.1·39-s − 4·43-s + 44/7·49-s + 5.47·57-s + 4·61-s − 5.31·67-s + 4.86·79-s − 0.358·81-s − 5.16·93-s + 4.86·97-s − 1.43·109-s + 6.27·111-s − 14.1·117-s + 0.00787·127-s − 8·129-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 88/7·147-s + 0.00671·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+1)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{140} \cdot 3^{20}\)
Sign: $1$
Analytic conductor: \(2.47357\times 10^{20}\)
Root analytic conductor: \(3.23469\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{140} \cdot 3^{20} ,\ ( \ : [1]^{20} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(33.98755983\)
\(L(\frac12)\) \(\approx\) \(33.98755983\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 2 p T + 2 p^{2} T^{2} - 10 p T^{3} + 65 T^{4} - 56 p T^{5} + 32 p^{2} T^{6} - 520 p T^{7} + 11278 T^{8} - 7148 p^{2} T^{9} + 2780 p^{4} T^{10} - 7148 p^{4} T^{11} + 11278 p^{4} T^{12} - 520 p^{7} T^{13} + 32 p^{10} T^{14} - 56 p^{11} T^{15} + 65 p^{12} T^{16} - 10 p^{15} T^{17} + 2 p^{18} T^{18} - 2 p^{19} T^{19} + p^{20} T^{20} \)
good5 \( 1 - 946 T^{4} + 556509 T^{8} + 5552424 p T^{12} - 71996170854 p T^{16} + 279323158314196 T^{20} - 71996170854 p^{9} T^{24} + 5552424 p^{17} T^{28} + 556509 p^{24} T^{32} - 946 p^{32} T^{36} + p^{40} T^{40} \)
7 \( ( 1 - 22 p T^{2} + 17009 T^{4} - 1179536 T^{6} + 71996590 T^{8} - 3527749420 T^{10} + 71996590 p^{4} T^{12} - 1179536 p^{8} T^{14} + 17009 p^{12} T^{16} - 22 p^{17} T^{18} + p^{20} T^{20} )^{2} \)
11 \( 1 - 17026 T^{4} - 443639331 T^{8} + 11991832514824 T^{12} + 4703618402526150 p T^{16} - \)\(36\!\cdots\!16\)\( T^{20} + 4703618402526150 p^{9} T^{24} + 11991832514824 p^{16} T^{28} - 443639331 p^{24} T^{32} - 17026 p^{32} T^{36} + p^{40} T^{40} \)
13 \( ( 1 + 46 T + 1058 T^{2} + 20510 T^{3} + 411997 T^{4} + 7447784 T^{5} + 117035288 T^{6} + 1803425192 T^{7} + 163445474 p^{2} T^{8} + 29706309732 p T^{9} + 5044801970700 T^{10} + 29706309732 p^{3} T^{11} + 163445474 p^{6} T^{12} + 1803425192 p^{6} T^{13} + 117035288 p^{8} T^{14} + 7447784 p^{10} T^{15} + 411997 p^{12} T^{16} + 20510 p^{14} T^{17} + 1058 p^{16} T^{18} + 46 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
17 \( ( 1 - 114 p T^{2} + 1855181 T^{4} - 1152381976 T^{6} + 513055082610 T^{8} - 170587207926956 T^{10} + 513055082610 p^{4} T^{12} - 1152381976 p^{8} T^{14} + 1855181 p^{12} T^{16} - 114 p^{17} T^{18} + p^{20} T^{20} )^{2} \)
19 \( ( 1 - 26 T + 338 T^{2} - 702 p T^{3} + 349281 T^{4} - 2446864 T^{5} + 34512608 T^{6} - 926396720 T^{7} - 22197634642 T^{8} + 577965275108 T^{9} - 4229043710052 T^{10} + 577965275108 p^{2} T^{11} - 22197634642 p^{4} T^{12} - 926396720 p^{6} T^{13} + 34512608 p^{8} T^{14} - 2446864 p^{10} T^{15} + 349281 p^{12} T^{16} - 702 p^{15} T^{17} + 338 p^{16} T^{18} - 26 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
23 \( ( 1 + 3054 T^{2} + 4762717 T^{4} + 4930394824 T^{6} + 3773391227074 T^{8} + 2245272418513300 T^{10} + 3773391227074 p^{4} T^{12} + 4930394824 p^{8} T^{14} + 4762717 p^{12} T^{16} + 3054 p^{16} T^{18} + p^{20} T^{20} )^{2} \)
29 \( 1 + 865038 T^{4} - 341999726179 T^{8} - 621091035708977976 T^{12} - \)\(34\!\cdots\!02\)\( T^{16} + \)\(22\!\cdots\!76\)\( T^{20} - \)\(34\!\cdots\!02\)\( p^{8} T^{24} - 621091035708977976 p^{16} T^{28} - 341999726179 p^{24} T^{32} + 865038 p^{32} T^{36} + p^{40} T^{40} \)
31 \( ( 1 + 20 T + 2055 T^{2} + 69364 T^{3} + 2243976 T^{4} + 102850448 T^{5} + 2243976 p^{2} T^{6} + 69364 p^{4} T^{7} + 2055 p^{6} T^{8} + 20 p^{8} T^{9} + p^{10} T^{10} )^{4} \)
37 \( ( 1 - 58 T + 1682 T^{2} + 50742 T^{3} - 603459 T^{4} - 139637784 T^{5} + 10401384792 T^{6} - 114472119576 T^{7} + 338413283634 T^{8} - 111361866221948 T^{9} + 16107397431686060 T^{10} - 111361866221948 p^{2} T^{11} + 338413283634 p^{4} T^{12} - 114472119576 p^{6} T^{13} + 10401384792 p^{8} T^{14} - 139637784 p^{10} T^{15} - 603459 p^{12} T^{16} + 50742 p^{14} T^{17} + 1682 p^{16} T^{18} - 58 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
41 \( ( 1 + 8166 T^{2} + 35165469 T^{4} + 106382596584 T^{6} + 246760101929730 T^{8} + 458809512185363300 T^{10} + 246760101929730 p^{4} T^{12} + 106382596584 p^{8} T^{14} + 35165469 p^{12} T^{16} + 8166 p^{16} T^{18} + p^{20} T^{20} )^{2} \)
43 \( ( 1 + 2 p T + 2 p^{2} T^{2} + 270774 T^{3} + 16192641 T^{4} + 356953072 T^{5} + 7476857312 T^{6} + 199821444560 T^{7} - 33578430535506 T^{8} - 2320167893729020 T^{9} - 78617269367427492 T^{10} - 2320167893729020 p^{2} T^{11} - 33578430535506 p^{4} T^{12} + 199821444560 p^{6} T^{13} + 7476857312 p^{8} T^{14} + 356953072 p^{10} T^{15} + 16192641 p^{12} T^{16} + 270774 p^{14} T^{17} + 2 p^{18} T^{18} + 2 p^{19} T^{19} + p^{20} T^{20} )^{2} \)
47 \( ( 1 - 17146 T^{2} + 138875501 T^{4} - 703714777016 T^{6} + 2481001995058130 T^{8} - 6375856842165200540 T^{10} + 2481001995058130 p^{4} T^{12} - 703714777016 p^{8} T^{14} + 138875501 p^{12} T^{16} - 17146 p^{16} T^{18} + p^{20} T^{20} )^{2} \)
53 \( 1 + 1437518 T^{4} + 119553426803037 T^{8} - \)\(89\!\cdots\!68\)\( T^{12} + \)\(36\!\cdots\!22\)\( T^{16} - \)\(10\!\cdots\!60\)\( T^{20} + \)\(36\!\cdots\!22\)\( p^{8} T^{24} - \)\(89\!\cdots\!68\)\( p^{16} T^{28} + 119553426803037 p^{24} T^{32} + 1437518 p^{32} T^{36} + p^{40} T^{40} \)
59 \( 1 - 19699682 T^{4} + 5882421086685 T^{8} + \)\(56\!\cdots\!08\)\( T^{12} + \)\(29\!\cdots\!74\)\( T^{16} - \)\(57\!\cdots\!12\)\( T^{20} + \)\(29\!\cdots\!74\)\( p^{8} T^{24} + \)\(56\!\cdots\!08\)\( p^{16} T^{28} + 5882421086685 p^{24} T^{32} - 19699682 p^{32} T^{36} + p^{40} T^{40} \)
61 \( ( 1 - 2 p T + 2 p^{2} T^{2} - 699306 T^{3} + 60909597 T^{4} - 2415315288 T^{5} + 85893685080 T^{6} - 3780099265368 T^{7} - 289655325423054 T^{8} + 34035914575403972 T^{9} - 1668196322377933396 T^{10} + 34035914575403972 p^{2} T^{11} - 289655325423054 p^{4} T^{12} - 3780099265368 p^{6} T^{13} + 85893685080 p^{8} T^{14} - 2415315288 p^{10} T^{15} + 60909597 p^{12} T^{16} - 699306 p^{14} T^{17} + 2 p^{18} T^{18} - 2 p^{19} T^{19} + p^{20} T^{20} )^{2} \)
67 \( ( 1 + 178 T + 15842 T^{2} + 1501482 T^{3} + 163953249 T^{4} + 13424405736 T^{5} + 919420948512 T^{6} + 73920183453528 T^{7} + 6211456955376366 T^{8} + 413849456030337652 T^{9} + 25618409953668575740 T^{10} + 413849456030337652 p^{2} T^{11} + 6211456955376366 p^{4} T^{12} + 73920183453528 p^{6} T^{13} + 919420948512 p^{8} T^{14} + 13424405736 p^{10} T^{15} + 163953249 p^{12} T^{16} + 1501482 p^{14} T^{17} + 15842 p^{16} T^{18} + 178 p^{18} T^{19} + p^{20} T^{20} )^{2} \)
71 \( ( 1 + 37534 T^{2} + 679811933 T^{4} + 7799021001352 T^{6} + 62618275556164866 T^{8} + \)\(36\!\cdots\!28\)\( T^{10} + 62618275556164866 p^{4} T^{12} + 7799021001352 p^{8} T^{14} + 679811933 p^{12} T^{16} + 37534 p^{16} T^{18} + p^{20} T^{20} )^{2} \)
73 \( ( 1 - 37130 T^{2} + 665990797 T^{4} - 7634966781176 T^{6} + 62349742704798482 T^{8} - \)\(38\!\cdots\!28\)\( T^{10} + 62349742704798482 p^{4} T^{12} - 7634966781176 p^{8} T^{14} + 665990797 p^{12} T^{16} - 37130 p^{16} T^{18} + p^{20} T^{20} )^{2} \)
79 \( ( 1 - 96 T + 27671 T^{2} - 2133500 T^{3} + 327107352 T^{4} - 19299334696 T^{5} + 327107352 p^{2} T^{6} - 2133500 p^{4} T^{7} + 27671 p^{6} T^{8} - 96 p^{8} T^{9} + p^{10} T^{10} )^{4} \)
83 \( 1 + 11186750 T^{4} - 558936874100067 T^{8} - \)\(49\!\cdots\!36\)\( T^{12} + \)\(20\!\cdots\!66\)\( T^{16} + \)\(44\!\cdots\!52\)\( T^{20} + \)\(20\!\cdots\!66\)\( p^{8} T^{24} - \)\(49\!\cdots\!36\)\( p^{16} T^{28} - 558936874100067 p^{24} T^{32} + 11186750 p^{32} T^{36} + p^{40} T^{40} \)
89 \( ( 1 + 29470 T^{2} + 451584989 T^{4} + 4107082352008 T^{6} + 26252370181550850 T^{8} + \)\(16\!\cdots\!64\)\( T^{10} + 26252370181550850 p^{4} T^{12} + 4107082352008 p^{8} T^{14} + 451584989 p^{12} T^{16} + 29470 p^{16} T^{18} + p^{20} T^{20} )^{2} \)
97 \( ( 1 - 118 T + 35265 T^{2} - 3292640 T^{3} + 583062270 T^{4} - 43725541204 T^{5} + 583062270 p^{2} T^{6} - 3292640 p^{4} T^{7} + 35265 p^{6} T^{8} - 118 p^{8} T^{9} + p^{10} T^{10} )^{4} \)
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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

−2.40466137976991044871459715391, −2.37899908781492606395560620987, −2.35912831202153136147952703107, −2.27173218779678413416540058141, −2.06476238451880831744310427836, −2.05557486707129818188285189151, −2.01049445344633146170850902611, −1.87666960785282079337315279073, −1.79526889661747492982981382900, −1.78961474901159798196075027766, −1.74189405630350123931959780258, −1.68366108747882099320568970592, −1.62783402359090038259092054609, −1.35416425487326844351342360959, −1.11158010955295498384716367260, −1.05439132120166771931396458118, −1.04000172437002276387750193555, −0.915345119695454790440431874652, −0.802539649357753360204362816719, −0.74243228896546572872097761396, −0.45618161345273212472290891794, −0.42102153555175260934399637179, −0.41774424184818042077058480175, −0.22463193524008575593133188587, −0.21447102864772696759487961378, 0.21447102864772696759487961378, 0.22463193524008575593133188587, 0.41774424184818042077058480175, 0.42102153555175260934399637179, 0.45618161345273212472290891794, 0.74243228896546572872097761396, 0.802539649357753360204362816719, 0.915345119695454790440431874652, 1.04000172437002276387750193555, 1.05439132120166771931396458118, 1.11158010955295498384716367260, 1.35416425487326844351342360959, 1.62783402359090038259092054609, 1.68366108747882099320568970592, 1.74189405630350123931959780258, 1.78961474901159798196075027766, 1.79526889661747492982981382900, 1.87666960785282079337315279073, 2.01049445344633146170850902611, 2.05557486707129818188285189151, 2.06476238451880831744310427836, 2.27173218779678413416540058141, 2.35912831202153136147952703107, 2.37899908781492606395560620987, 2.40466137976991044871459715391

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

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