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Properties

Label 12-3e36-1.1-c1e6-0-10
Degree 1212
Conductor 1.501×10171.501\times 10^{17}
Sign 11
Analytic cond. 38907.038907.0
Root an. cond. 2.412692.41269
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 6·5-s + 9·7-s − 9·8-s + 18·10-s + 3·11-s + 9·13-s + 27·14-s − 9·16-s + 9·17-s + 3·19-s + 9·22-s + 15·23-s + 27·25-s + 27·26-s − 15·29-s + 9·31-s + 3·32-s + 27·34-s + 54·35-s + 3·37-s + 9·38-s − 54·40-s − 3·41-s + 9·43-s + 45·46-s − 15·47-s + ⋯
L(s)  = 1  + 2.12·2-s + 2.68·5-s + 3.40·7-s − 3.18·8-s + 5.69·10-s + 0.904·11-s + 2.49·13-s + 7.21·14-s − 9/4·16-s + 2.18·17-s + 0.688·19-s + 1.91·22-s + 3.12·23-s + 27/5·25-s + 5.29·26-s − 2.78·29-s + 1.61·31-s + 0.530·32-s + 4.63·34-s + 9.12·35-s + 0.493·37-s + 1.45·38-s − 8.53·40-s − 0.468·41-s + 1.37·43-s + 6.63·46-s − 2.18·47-s + ⋯

Functional equation

Λ(s)=((336)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((336)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 3363^{36}
Sign: 11
Analytic conductor: 38907.038907.0
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 336, ( :[1/2]6), 1)(12,\ 3^{36} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 52.7572987252.75729872
L(12)L(\frac12) ≈\approx 52.7572987252.75729872
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
good2 1−3T+9T2−9pT3+9p2T4−57T5+91T6−57pT7+9p4T8−9p4T9+9p4T10−3p5T11+p6T12 1 - 3 T + 9 T^{2} - 9 p T^{3} + 9 p^{2} T^{4} - 57 T^{5} + 91 T^{6} - 57 p T^{7} + 9 p^{4} T^{8} - 9 p^{4} T^{9} + 9 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
5 1−6T+9T2+27T3−99T4−33T5+514T6−33pT7−99p2T8+27p3T9+9p4T10−6p5T11+p6T12 1 - 6 T + 9 T^{2} + 27 T^{3} - 99 T^{4} - 33 T^{5} + 514 T^{6} - 33 p T^{7} - 99 p^{2} T^{8} + 27 p^{3} T^{9} + 9 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
7 1−9T+36T2−10pT3+513T5−1923T6+513pT7−10p4T9+36p4T10−9p5T11+p6T12 1 - 9 T + 36 T^{2} - 10 p T^{3} + 513 T^{5} - 1923 T^{6} + 513 p T^{7} - 10 p^{4} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
11 1−3T+36T2−126T3+684T4−2451T5+8677T6−2451pT7+684p2T8−126p3T9+36p4T10−3p5T11+p6T12 1 - 3 T + 36 T^{2} - 126 T^{3} + 684 T^{4} - 2451 T^{5} + 8677 T^{6} - 2451 p T^{7} + 684 p^{2} T^{8} - 126 p^{3} T^{9} + 36 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
13 1−9T+36T2−4pT3−162T4+1917T5−8889T6+1917pT7−162p2T8−4p4T9+36p4T10−9p5T11+p6T12 1 - 9 T + 36 T^{2} - 4 p T^{3} - 162 T^{4} + 1917 T^{5} - 8889 T^{6} + 1917 p T^{7} - 162 p^{2} T^{8} - 4 p^{4} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
17 (1−3T−8T2−3pT3+p2T4)3 ( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{3}
19 1−3T−24T2+131T3+117T4−1116T5+3003T6−1116pT7+117p2T8+131p3T9−24p4T10−3p5T11+p6T12 1 - 3 T - 24 T^{2} + 131 T^{3} + 117 T^{4} - 1116 T^{5} + 3003 T^{6} - 1116 p T^{7} + 117 p^{2} T^{8} + 131 p^{3} T^{9} - 24 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
23 1−15T+90T2−162T3−1017T4+10101T5−54485T6+10101pT7−1017p2T8−162p3T9+90p4T10−15p5T11+p6T12 1 - 15 T + 90 T^{2} - 162 T^{3} - 1017 T^{4} + 10101 T^{5} - 54485 T^{6} + 10101 p T^{7} - 1017 p^{2} T^{8} - 162 p^{3} T^{9} + 90 p^{4} T^{10} - 15 p^{5} T^{11} + p^{6} T^{12}
29 1+15T+144T2+1188T3+8163T4+49317T5+277705T6+49317pT7+8163p2T8+1188p3T9+144p4T10+15p5T11+p6T12 1 + 15 T + 144 T^{2} + 1188 T^{3} + 8163 T^{4} + 49317 T^{5} + 277705 T^{6} + 49317 p T^{7} + 8163 p^{2} T^{8} + 1188 p^{3} T^{9} + 144 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12}
31 1−9T+36T2+164T3−27pT4−1377T5+53805T6−1377pT7−27p3T8+164p3T9+36p4T10−9p5T11+p6T12 1 - 9 T + 36 T^{2} + 164 T^{3} - 27 p T^{4} - 1377 T^{5} + 53805 T^{6} - 1377 p T^{7} - 27 p^{3} T^{8} + 164 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
37 1−3T−78T2+5pT3+3681T4−4194T5−141339T6−4194pT7+3681p2T8+5p4T9−78p4T10−3p5T11+p6T12 1 - 3 T - 78 T^{2} + 5 p T^{3} + 3681 T^{4} - 4194 T^{5} - 141339 T^{6} - 4194 p T^{7} + 3681 p^{2} T^{8} + 5 p^{4} T^{9} - 78 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
41 1+3T+36T2−72T3+18pT4+1119T5+93799T6+1119pT7+18p3T8−72p3T9+36p4T10+3p5T11+p6T12 1 + 3 T + 36 T^{2} - 72 T^{3} + 18 p T^{4} + 1119 T^{5} + 93799 T^{6} + 1119 p T^{7} + 18 p^{3} T^{8} - 72 p^{3} T^{9} + 36 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}
43 1−9T+36T2+308T3−1485T4+2403T5+82749T6+2403pT7−1485p2T8+308p3T9+36p4T10−9p5T11+p6T12 1 - 9 T + 36 T^{2} + 308 T^{3} - 1485 T^{4} + 2403 T^{5} + 82749 T^{6} + 2403 p T^{7} - 1485 p^{2} T^{8} + 308 p^{3} T^{9} + 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
47 1+15T+90T2+126T3+765T4+40155T5+461953T6+40155pT7+765p2T8+126p3T9+90p4T10+15p5T11+p6T12 1 + 15 T + 90 T^{2} + 126 T^{3} + 765 T^{4} + 40155 T^{5} + 461953 T^{6} + 40155 p T^{7} + 765 p^{2} T^{8} + 126 p^{3} T^{9} + 90 p^{4} T^{10} + 15 p^{5} T^{11} + p^{6} T^{12}
53 (1+18T+240T2+1989T3+240pT4+18p2T5+p3T6)2 ( 1 + 18 T + 240 T^{2} + 1989 T^{3} + 240 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2}
59 1−6T+36T2−999T3+4329T4−34053T5+664957T6−34053pT7+4329p2T8−999p3T9+36p4T10−6p5T11+p6T12 1 - 6 T + 36 T^{2} - 999 T^{3} + 4329 T^{4} - 34053 T^{5} + 664957 T^{6} - 34053 p T^{7} + 4329 p^{2} T^{8} - 999 p^{3} T^{9} + 36 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
61 1−18T+153T2−745T3−3915T4+107703T5−1034862T6+107703pT7−3915p2T8−745p3T9+153p4T10−18p5T11+p6T12 1 - 18 T + 153 T^{2} - 745 T^{3} - 3915 T^{4} + 107703 T^{5} - 1034862 T^{6} + 107703 p T^{7} - 3915 p^{2} T^{8} - 745 p^{3} T^{9} + 153 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12}
67 1+9T+126T2+758T3+8937T4+25569T5+327585T6+25569pT7+8937p2T8+758p3T9+126p4T10+9p5T11+p6T12 1 + 9 T + 126 T^{2} + 758 T^{3} + 8937 T^{4} + 25569 T^{5} + 327585 T^{6} + 25569 p T^{7} + 8937 p^{2} T^{8} + 758 p^{3} T^{9} + 126 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12}
71 1+9T+30T2−99T3−3531T4−5580T5+200671T6−5580pT7−3531p2T8−99p3T9+30p4T10+9p5T11+p6T12 1 + 9 T + 30 T^{2} - 99 T^{3} - 3531 T^{4} - 5580 T^{5} + 200671 T^{6} - 5580 p T^{7} - 3531 p^{2} T^{8} - 99 p^{3} T^{9} + 30 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12}
73 1+6T−114T2−58T3+8676T4−27576T5−826869T6−27576pT7+8676p2T8−58p3T9−114p4T10+6p5T11+p6T12 1 + 6 T - 114 T^{2} - 58 T^{3} + 8676 T^{4} - 27576 T^{5} - 826869 T^{6} - 27576 p T^{7} + 8676 p^{2} T^{8} - 58 p^{3} T^{9} - 114 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12}
79 1−9T+144T2−1096T3+19413T4−150957T5+1716951T6−150957pT7+19413p2T8−1096p3T9+144p4T10−9p5T11+p6T12 1 - 9 T + 144 T^{2} - 1096 T^{3} + 19413 T^{4} - 150957 T^{5} + 1716951 T^{6} - 150957 p T^{7} + 19413 p^{2} T^{8} - 1096 p^{3} T^{9} + 144 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
83 1−21T+198T2−1386T3+1953T4+125691T5−1739735T6+125691pT7+1953p2T8−1386p3T9+198p4T10−21p5T11+p6T12 1 - 21 T + 198 T^{2} - 1386 T^{3} + 1953 T^{4} + 125691 T^{5} - 1739735 T^{6} + 125691 p T^{7} + 1953 p^{2} T^{8} - 1386 p^{3} T^{9} + 198 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12}
89 1−78T2−1998T3−858T4+77922T5+1866463T6+77922pT7−858p2T8−1998p3T9−78p4T10+p6T12 1 - 78 T^{2} - 1998 T^{3} - 858 T^{4} + 77922 T^{5} + 1866463 T^{6} + 77922 p T^{7} - 858 p^{2} T^{8} - 1998 p^{3} T^{9} - 78 p^{4} T^{10} + p^{6} T^{12}
97 1−36T+576T2−4498T3−4860T4+578988T5−7833165T6+578988pT7−4860p2T8−4498p3T9+576p4T10−36p5T11+p6T12 1 - 36 T + 576 T^{2} - 4498 T^{3} - 4860 T^{4} + 578988 T^{5} - 7833165 T^{6} + 578988 p T^{7} - 4860 p^{2} T^{8} - 4498 p^{3} T^{9} + 576 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.54030650392550113103358919537, −5.38271481210261889698086677718, −5.07172410766409037885572274729, −4.97162961189259660385637688532, −4.85536853846760588303286308595, −4.83795533021789518208528383653, −4.82526419880881362444051952003, −4.34748643567407616672423956551, −4.29422544905455761499982613800, −4.14684791338315242107953857003, −3.82905360015540609018045200664, −3.68136969772610726992969723728, −3.63018272960433066693562651135, −3.27037687333684196147575573079, −3.14360952021820085942727265727, −2.89814604064344455828142901284, −2.65832921717516611532379786473, −2.60277388037505361012481497282, −2.07359958268742720250439342234, −1.68475625065057087344874778634, −1.59003019231981783883332441344, −1.43703667937789102383478970181, −1.32792992481616282169851449775, −1.00415111810951139295098173432, −0.911335230770448958832362849511, 0.911335230770448958832362849511, 1.00415111810951139295098173432, 1.32792992481616282169851449775, 1.43703667937789102383478970181, 1.59003019231981783883332441344, 1.68475625065057087344874778634, 2.07359958268742720250439342234, 2.60277388037505361012481497282, 2.65832921717516611532379786473, 2.89814604064344455828142901284, 3.14360952021820085942727265727, 3.27037687333684196147575573079, 3.63018272960433066693562651135, 3.68136969772610726992969723728, 3.82905360015540609018045200664, 4.14684791338315242107953857003, 4.29422544905455761499982613800, 4.34748643567407616672423956551, 4.82526419880881362444051952003, 4.83795533021789518208528383653, 4.85536853846760588303286308595, 4.97162961189259660385637688532, 5.07172410766409037885572274729, 5.38271481210261889698086677718, 5.54030650392550113103358919537

Graph of the ZZ-function along the critical line

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