Properties

Label 12-91e12-1.1-c1e6-0-0
Degree 1212
Conductor 3.225×10233.225\times 10^{23}
Sign 11
Analytic cond. 8.35909×10108.35909\times 10^{10}
Root an. cond. 8.131678.13167
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s − 2·4-s − 5-s + 2·6-s − 7·8-s − 10·9-s − 2·10-s + 4·11-s − 2·12-s − 15-s − 4·16-s + 5·17-s − 20·18-s + 19-s + 2·20-s + 8·22-s + 23-s − 7·24-s − 18·25-s − 10·27-s − 3·29-s − 2·30-s − 16·31-s + 4·32-s + 4·33-s + 10·34-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s − 4-s − 0.447·5-s + 0.816·6-s − 2.47·8-s − 3.33·9-s − 0.632·10-s + 1.20·11-s − 0.577·12-s − 0.258·15-s − 16-s + 1.21·17-s − 4.71·18-s + 0.229·19-s + 0.447·20-s + 1.70·22-s + 0.208·23-s − 1.42·24-s − 3.59·25-s − 1.92·27-s − 0.557·29-s − 0.365·30-s − 2.87·31-s + 0.707·32-s + 0.696·33-s + 1.71·34-s + ⋯

Functional equation

Λ(s)=((7121312)s/2ΓC(s)6L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((7121312)s/2ΓC(s+1/2)6L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 71213127^{12} \cdot 13^{12}
Sign: 11
Analytic conductor: 8.35909×10108.35909\times 10^{10}
Root analytic conductor: 8.131678.13167
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 7121312, ( :[1/2]6), 1)(12,\ 7^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) \approx 1.0907041651.090704165
L(12)L(\frac12) \approx 1.0907041651.090704165
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad7 1 1
13 1 1
good2 1pT+3pT29T3+5p2T47p2T5+51T67p3T7+5p4T89p3T9+3p5T10p6T11+p6T12 1 - p T + 3 p T^{2} - 9 T^{3} + 5 p^{2} T^{4} - 7 p^{2} T^{5} + 51 T^{6} - 7 p^{3} T^{7} + 5 p^{4} T^{8} - 9 p^{3} T^{9} + 3 p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12}
3 1T+11T211T3+19pT455T5+197T655pT7+19p3T811p3T9+11p4T10p5T11+p6T12 1 - T + 11 T^{2} - 11 T^{3} + 19 p T^{4} - 55 T^{5} + 197 T^{6} - 55 p T^{7} + 19 p^{3} T^{8} - 11 p^{3} T^{9} + 11 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12}
5 1+T+19T2+7T3+161T43T5+913T63pT7+161p2T8+7p3T9+19p4T10+p5T11+p6T12 1 + T + 19 T^{2} + 7 T^{3} + 161 T^{4} - 3 T^{5} + 913 T^{6} - 3 p T^{7} + 161 p^{2} T^{8} + 7 p^{3} T^{9} + 19 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12}
11 14T+45T2144T3+972T42539T5+13237T62539pT7+972p2T8144p3T9+45p4T104p5T11+p6T12 1 - 4 T + 45 T^{2} - 144 T^{3} + 972 T^{4} - 2539 T^{5} + 13237 T^{6} - 2539 p T^{7} + 972 p^{2} T^{8} - 144 p^{3} T^{9} + 45 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
17 15T+90T2411T3+3539T413744T5+78123T613744pT7+3539p2T8411p3T9+90p4T105p5T11+p6T12 1 - 5 T + 90 T^{2} - 411 T^{3} + 3539 T^{4} - 13744 T^{5} + 78123 T^{6} - 13744 p T^{7} + 3539 p^{2} T^{8} - 411 p^{3} T^{9} + 90 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12}
19 1T+50T2+16T3+1180T4+1175T5+23331T6+1175pT7+1180p2T8+16p3T9+50p4T10p5T11+p6T12 1 - T + 50 T^{2} + 16 T^{3} + 1180 T^{4} + 1175 T^{5} + 23331 T^{6} + 1175 p T^{7} + 1180 p^{2} T^{8} + 16 p^{3} T^{9} + 50 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12}
23 1T+32T252T3+1214T41383T5+21935T61383pT7+1214p2T852p3T9+32p4T10p5T11+p6T12 1 - T + 32 T^{2} - 52 T^{3} + 1214 T^{4} - 1383 T^{5} + 21935 T^{6} - 1383 p T^{7} + 1214 p^{2} T^{8} - 52 p^{3} T^{9} + 32 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12}
29 1+3T+96T2+191T3+4061T4+5126T5+122643T6+5126pT7+4061p2T8+191p3T9+96p4T10+3p5T11+p6T12 1 + 3 T + 96 T^{2} + 191 T^{3} + 4061 T^{4} + 5126 T^{5} + 122643 T^{6} + 5126 p T^{7} + 4061 p^{2} T^{8} + 191 p^{3} T^{9} + 96 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}
31 1+16T+236T2+2185T3+18573T4+122325T5+755039T6+122325pT7+18573p2T8+2185p3T9+236p4T10+16p5T11+p6T12 1 + 16 T + 236 T^{2} + 2185 T^{3} + 18573 T^{4} + 122325 T^{5} + 755039 T^{6} + 122325 p T^{7} + 18573 p^{2} T^{8} + 2185 p^{3} T^{9} + 236 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12}
37 1+13T+184T2+1054T3+7158T4+10573T5+113729T6+10573pT7+7158p2T8+1054p3T9+184p4T10+13p5T11+p6T12 1 + 13 T + 184 T^{2} + 1054 T^{3} + 7158 T^{4} + 10573 T^{5} + 113729 T^{6} + 10573 p T^{7} + 7158 p^{2} T^{8} + 1054 p^{3} T^{9} + 184 p^{4} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12}
41 18T+225T21362T3+21488T4101725T5+1145451T6101725pT7+21488p2T81362p3T9+225p4T108p5T11+p6T12 1 - 8 T + 225 T^{2} - 1362 T^{3} + 21488 T^{4} - 101725 T^{5} + 1145451 T^{6} - 101725 p T^{7} + 21488 p^{2} T^{8} - 1362 p^{3} T^{9} + 225 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12}
43 111T+259T22099T3+27622T4170696T5+1576761T6170696pT7+27622p2T82099p3T9+259p4T1011p5T11+p6T12 1 - 11 T + 259 T^{2} - 2099 T^{3} + 27622 T^{4} - 170696 T^{5} + 1576761 T^{6} - 170696 p T^{7} + 27622 p^{2} T^{8} - 2099 p^{3} T^{9} + 259 p^{4} T^{10} - 11 p^{5} T^{11} + p^{6} T^{12}
47 1T+105T2147T3+5543T43359T5+246951T63359pT7+5543p2T8147p3T9+105p4T10p5T11+p6T12 1 - T + 105 T^{2} - 147 T^{3} + 5543 T^{4} - 3359 T^{5} + 246951 T^{6} - 3359 p T^{7} + 5543 p^{2} T^{8} - 147 p^{3} T^{9} + 105 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12}
53 12T+218T2344T3+22040T426940T5+1409201T626940pT7+22040p2T8344p3T9+218p4T102p5T11+p6T12 1 - 2 T + 218 T^{2} - 344 T^{3} + 22040 T^{4} - 26940 T^{5} + 1409201 T^{6} - 26940 p T^{7} + 22040 p^{2} T^{8} - 344 p^{3} T^{9} + 218 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
59 1+13T+5pT2+2839T3+38957T4+294699T5+2963017T6+294699pT7+38957p2T8+2839p3T9+5p5T10+13p5T11+p6T12 1 + 13 T + 5 p T^{2} + 2839 T^{3} + 38957 T^{4} + 294699 T^{5} + 2963017 T^{6} + 294699 p T^{7} + 38957 p^{2} T^{8} + 2839 p^{3} T^{9} + 5 p^{5} T^{10} + 13 p^{5} T^{11} + p^{6} T^{12}
61 1+5T+165T2+599T3+15743T4+53393T5+1179159T6+53393pT7+15743p2T8+599p3T9+165p4T10+5p5T11+p6T12 1 + 5 T + 165 T^{2} + 599 T^{3} + 15743 T^{4} + 53393 T^{5} + 1179159 T^{6} + 53393 p T^{7} + 15743 p^{2} T^{8} + 599 p^{3} T^{9} + 165 p^{4} T^{10} + 5 p^{5} T^{11} + p^{6} T^{12}
67 1+11T+296T2+2796T3+41197T4+326168T5+3447813T6+326168pT7+41197p2T8+2796p3T9+296p4T10+11p5T11+p6T12 1 + 11 T + 296 T^{2} + 2796 T^{3} + 41197 T^{4} + 326168 T^{5} + 3447813 T^{6} + 326168 p T^{7} + 41197 p^{2} T^{8} + 2796 p^{3} T^{9} + 296 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12}
71 16T+285T214pT3+35468T474185T5+2901951T674185pT7+35468p2T814p4T9+285p4T106p5T11+p6T12 1 - 6 T + 285 T^{2} - 14 p T^{3} + 35468 T^{4} - 74185 T^{5} + 2901951 T^{6} - 74185 p T^{7} + 35468 p^{2} T^{8} - 14 p^{4} T^{9} + 285 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12}
73 130T+676T210699T3+141027T41519265T5+14149139T61519265pT7+141027p2T810699p3T9+676p4T1030p5T11+p6T12 1 - 30 T + 676 T^{2} - 10699 T^{3} + 141027 T^{4} - 1519265 T^{5} + 14149139 T^{6} - 1519265 p T^{7} + 141027 p^{2} T^{8} - 10699 p^{3} T^{9} + 676 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12}
79 1+7T+326T2+2455T3+54096T4+344443T5+5474643T6+344443pT7+54096p2T8+2455p3T9+326p4T10+7p5T11+p6T12 1 + 7 T + 326 T^{2} + 2455 T^{3} + 54096 T^{4} + 344443 T^{5} + 5474643 T^{6} + 344443 p T^{7} + 54096 p^{2} T^{8} + 2455 p^{3} T^{9} + 326 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12}
83 1+27T+656T2+10802T3+153994T4+1760871T5+17670883T6+1760871pT7+153994p2T8+10802p3T9+656p4T10+27p5T11+p6T12 1 + 27 T + 656 T^{2} + 10802 T^{3} + 153994 T^{4} + 1760871 T^{5} + 17670883 T^{6} + 1760871 p T^{7} + 153994 p^{2} T^{8} + 10802 p^{3} T^{9} + 656 p^{4} T^{10} + 27 p^{5} T^{11} + p^{6} T^{12}
89 1+4T+167T2+1648T3+21035T4+202110T5+2204075T6+202110pT7+21035p2T8+1648p3T9+167p4T10+4p5T11+p6T12 1 + 4 T + 167 T^{2} + 1648 T^{3} + 21035 T^{4} + 202110 T^{5} + 2204075 T^{6} + 202110 p T^{7} + 21035 p^{2} T^{8} + 1648 p^{3} T^{9} + 167 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
97 135T+947T218161T3+281670T43629766T5+38644781T63629766pT7+281670p2T818161p3T9+947p4T1035p5T11+p6T12 1 - 35 T + 947 T^{2} - 18161 T^{3} + 281670 T^{4} - 3629766 T^{5} + 38644781 T^{6} - 3629766 p T^{7} + 281670 p^{2} T^{8} - 18161 p^{3} T^{9} + 947 p^{4} T^{10} - 35 p^{5} T^{11} + p^{6} T^{12}
show more
show less
   L(s)=p j=112(1αj,pps)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

−4.02301834815495183224958416038, −3.68241364426194685139385720080, −3.64491283110275752117144736944, −3.61304943038765930289590630884, −3.53835350614457093369441444045, −3.46767223481295891361957026408, −3.29145931385397189884827768778, −3.08016723100250928398057604476, −2.89511292394892601136779838335, −2.83903014321783801968820330978, −2.76329155787451344306404330881, −2.65294708378062721644923515938, −2.23851562029172358502785151207, −2.14240052526736777498970592628, −2.03422250890345436126006244357, −1.96261037083161603607083263733, −1.86246597012376464590779029227, −1.49050829122657866389018035259, −1.42483110475111036611536594729, −1.36699431739397690870270681519, −0.68355653587700443792888520791, −0.65785245355063660305618623073, −0.48406870277325265509681665042, −0.42658562472398187631631534260, −0.10710580178647573481117557401, 0.10710580178647573481117557401, 0.42658562472398187631631534260, 0.48406870277325265509681665042, 0.65785245355063660305618623073, 0.68355653587700443792888520791, 1.36699431739397690870270681519, 1.42483110475111036611536594729, 1.49050829122657866389018035259, 1.86246597012376464590779029227, 1.96261037083161603607083263733, 2.03422250890345436126006244357, 2.14240052526736777498970592628, 2.23851562029172358502785151207, 2.65294708378062721644923515938, 2.76329155787451344306404330881, 2.83903014321783801968820330978, 2.89511292394892601136779838335, 3.08016723100250928398057604476, 3.29145931385397189884827768778, 3.46767223481295891361957026408, 3.53835350614457093369441444045, 3.61304943038765930289590630884, 3.64491283110275752117144736944, 3.68241364426194685139385720080, 4.02301834815495183224958416038

Graph of the ZZ-function along the critical line

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