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Properties

Label 12-120e6-1.1-c1e6-0-0
Degree 1212
Conductor 2.986×10122.986\times 10^{12}
Sign 11
Analytic cond. 0.7740160.774016
Root an. cond. 0.9788790.978879
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  + 2-s − 6·3-s + 4-s − 6·6-s + 8-s + 21·9-s − 6·12-s + 8·13-s + 16-s + 21·18-s − 6·24-s + 25-s + 8·26-s − 56·27-s − 16·31-s + 5·32-s + 21·36-s + 16·37-s − 48·39-s − 4·41-s − 6·48-s + 18·49-s + 50-s + 8·52-s − 24·53-s − 56·54-s − 16·62-s + ⋯
L(s)  = 1  + 0.707·2-s − 3.46·3-s + 1/2·4-s − 2.44·6-s + 0.353·8-s + 7·9-s − 1.73·12-s + 2.21·13-s + 1/4·16-s + 4.94·18-s − 1.22·24-s + 1/5·25-s + 1.56·26-s − 10.7·27-s − 2.87·31-s + 0.883·32-s + 7/2·36-s + 2.63·37-s − 7.68·39-s − 0.624·41-s − 0.866·48-s + 18/7·49-s + 0.141·50-s + 1.10·52-s − 3.29·53-s − 7.62·54-s − 2.03·62-s + ⋯

Functional equation

Λ(s)=((218⋅36⋅56)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((218⋅36⋅56)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 218⋅36⋅562^{18} \cdot 3^{6} \cdot 5^{6}
Sign: 11
Analytic conductor: 0.7740160.774016
Root analytic conductor: 0.9788790.978879
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 218⋅36⋅56, ( :[1/2]6), 1)(12,\ 2^{18} \cdot 3^{6} \cdot 5^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.61964239100.6196423910
L(12)L(\frac12) ≈\approx 0.61964239100.6196423910
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)
bad2 1−T−p2T5+p3T6 1 - T - p^{2} T^{5} + p^{3} T^{6}
3 (1+T)6 ( 1 + T )^{6}
5 1−T2+8T3−pT4+p3T6 1 - T^{2} + 8 T^{3} - p T^{4} + p^{3} T^{6}
good7 1−18T2+191T4−1532T6+191p2T8−18p4T10+p6T12 1 - 18 T^{2} + 191 T^{4} - 1532 T^{6} + 191 p^{2} T^{8} - 18 p^{4} T^{10} + p^{6} T^{12}
11 1−34T2+503T4−5436T6+503p2T8−34p4T10+p6T12 1 - 34 T^{2} + 503 T^{4} - 5436 T^{6} + 503 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12}
13 (1−4T+23T2−48T3+23pT4−4p2T5+p3T6)2 ( 1 - 4 T + 23 T^{2} - 48 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2}
17 1−66T2+2255T4−47324T6+2255p2T8−66p4T10+p6T12 1 - 66 T^{2} + 2255 T^{4} - 47324 T^{6} + 2255 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12}
19 1−54T2+1367T4−25652T6+1367p2T8−54p4T10+p6T12 1 - 54 T^{2} + 1367 T^{4} - 25652 T^{6} + 1367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12}
23 1−2pT2+1775T4−40932T6+1775p2T8−2p5T10+p6T12 1 - 2 p T^{2} + 1775 T^{4} - 40932 T^{6} + 1775 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12}
29 1−66T2+3207T4−111228T6+3207p2T8−66p4T10+p6T12 1 - 66 T^{2} + 3207 T^{4} - 111228 T^{6} + 3207 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12}
31 (1+8T+89T2+432T3+89pT4+8p2T5+p3T6)2 ( 1 + 8 T + 89 T^{2} + 432 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
37 (1−8T+3pT2−584T3+3p2T4−8p2T5+p3T6)2 ( 1 - 8 T + 3 p T^{2} - 584 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
41 (1+2T+23T2+220T3+23pT4+2p2T5+p3T6)2 ( 1 + 2 T + 23 T^{2} + 220 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
43 (1+65T2−64T3+65pT4+p3T6)2 ( 1 + 65 T^{2} - 64 T^{3} + 65 p T^{4} + p^{3} T^{6} )^{2}
47 1−222T2+22367T4−1328324T6+22367p2T8−222p4T10+p6T12 1 - 222 T^{2} + 22367 T^{4} - 1328324 T^{6} + 22367 p^{2} T^{8} - 222 p^{4} T^{10} + p^{6} T^{12}
53 (1+12T+191T2+1264T3+191pT4+12p2T5+p3T6)2 ( 1 + 12 T + 191 T^{2} + 1264 T^{3} + 191 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} )^{2}
59 1−178T2+20567T4−1418652T6+20567p2T8−178p4T10+p6T12 1 - 178 T^{2} + 20567 T^{4} - 1418652 T^{6} + 20567 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12}
61 1−190T2+20039T4−1419204T6+20039p2T8−190p4T10+p6T12 1 - 190 T^{2} + 20039 T^{4} - 1419204 T^{6} + 20039 p^{2} T^{8} - 190 p^{4} T^{10} + p^{6} T^{12}
67 (1+137T2+64T3+137pT4+p3T6)2 ( 1 + 137 T^{2} + 64 T^{3} + 137 p T^{4} + p^{3} T^{6} )^{2}
71 (1−8T+133T2−1008T3+133pT4−8p2T5+p3T6)2 ( 1 - 8 T + 133 T^{2} - 1008 T^{3} + 133 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1−54T2+2367T4−531700T6+2367p2T8−54p4T10+p6T12 1 - 54 T^{2} + 2367 T^{4} - 531700 T^{6} + 2367 p^{2} T^{8} - 54 p^{4} T^{10} + p^{6} T^{12}
79 (1−8T+233T2−1200T3+233pT4−8p2T5+p3T6)2 ( 1 - 8 T + 233 T^{2} - 1200 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
83 (1−8T+185T2−880T3+185pT4−8p2T5+p3T6)2 ( 1 - 8 T + 185 T^{2} - 880 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
89 (1+10T+103T2+396T3+103pT4+10p2T5+p3T6)2 ( 1 + 10 T + 103 T^{2} + 396 T^{3} + 103 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2}
97 1−246T2+39183T4−4535476T6+39183p2T8−246p4T10+p6T12 1 - 246 T^{2} + 39183 T^{4} - 4535476 T^{6} + 39183 p^{2} T^{8} - 246 p^{4} T^{10} + 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

−7.41723458285377844873052520256, −7.23918524644620620596748114809, −7.05349848888652327567376920367, −6.86981232137470216174228105715, −6.63721449314990582705857205057, −6.22691155196623033128677063676, −6.15148969379611355540884927692, −6.14296126560268722196977971321, −6.07284318511331767459595709353, −5.69149156033938695983616953089, −5.42270100349333544655125648144, −5.33880860225500696184261446878, −5.06683614032494353272679440030, −4.87299565805114013270565354649, −4.41879866932439474026769339950, −4.39508971616457155186481528911, −4.22098229968495723923824090517, −3.74240884645171283901061950817, −3.48817147279504785598946044861, −3.46030293882061327036993626990, −2.87615994388096863389644470442, −2.04128664062871564010657097566, −2.03227801394467690408670626568, −1.24970307222334753202882376774, −0.884039741002023211256200436812, 0.884039741002023211256200436812, 1.24970307222334753202882376774, 2.03227801394467690408670626568, 2.04128664062871564010657097566, 2.87615994388096863389644470442, 3.46030293882061327036993626990, 3.48817147279504785598946044861, 3.74240884645171283901061950817, 4.22098229968495723923824090517, 4.39508971616457155186481528911, 4.41879866932439474026769339950, 4.87299565805114013270565354649, 5.06683614032494353272679440030, 5.33880860225500696184261446878, 5.42270100349333544655125648144, 5.69149156033938695983616953089, 6.07284318511331767459595709353, 6.14296126560268722196977971321, 6.15148969379611355540884927692, 6.22691155196623033128677063676, 6.63721449314990582705857205057, 6.86981232137470216174228105715, 7.05349848888652327567376920367, 7.23918524644620620596748114809, 7.41723458285377844873052520256

Graph of the ZZ-function along the critical line

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