↑

Properties

Label 12-637e6-1.1-c1e6-0-2
Degree 1212
Conductor 6.681×10166.681\times 10^{16}
Sign 11
Analytic cond. 17318.017318.0
Root an. cond. 2.255322.25532
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-s − 2·3-s + 2·4-s + 2·5-s + 2·6-s − 8-s + 3·9-s − 2·10-s − 2·11-s − 4·12-s − 6·13-s − 4·15-s + 3·16-s + 4·17-s − 3·18-s − 4·19-s + 4·20-s + 2·22-s − 10·23-s + 2·24-s + 12·25-s + 6·26-s − 2·27-s + 48·29-s + 4·30-s − 4·31-s − 2·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 4-s + 0.894·5-s + 0.816·6-s − 0.353·8-s + 9-s − 0.632·10-s − 0.603·11-s − 1.15·12-s − 1.66·13-s − 1.03·15-s + 3/4·16-s + 0.970·17-s − 0.707·18-s − 0.917·19-s + 0.894·20-s + 0.426·22-s − 2.08·23-s + 0.408·24-s + 12/5·25-s + 1.17·26-s − 0.384·27-s + 8.91·29-s + 0.730·30-s − 0.718·31-s − 0.353·32-s + ⋯

Functional equation

Λ(s)=((712⋅136)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((712⋅136)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 712⋅1367^{12} \cdot 13^{6}
Sign: 11
Analytic conductor: 17318.017318.0
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 712⋅136, ( :[1/2]6), 1)(12,\ 7^{12} \cdot 13^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 3.4315865023.431586502
L(12)L(\frac12) ≈\approx 3.4315865023.431586502
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)
bad7 1 1
13 (1+T)6 ( 1 + T )^{6}
good2 1+T−T2−pT3−pT4+p3T6−p3T8−p4T9−p4T10+p5T11+p6T12 1 + T - T^{2} - p T^{3} - p T^{4} + p^{3} T^{6} - p^{3} T^{8} - p^{4} T^{9} - p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12}
3 1+2T+T2−2T3−8T4−2pT5+7T6−2p2T7−8p2T8−2p3T9+p4T10+2p5T11+p6T12 1 + 2 T + T^{2} - 2 T^{3} - 8 T^{4} - 2 p T^{5} + 7 T^{6} - 2 p^{2} T^{7} - 8 p^{2} T^{8} - 2 p^{3} T^{9} + p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
5 1−2T−8T2+12T3+48T4−26T5−246T6−26pT7+48p2T8+12p3T9−8p4T10−2p5T11+p6T12 1 - 2 T - 8 T^{2} + 12 T^{3} + 48 T^{4} - 26 T^{5} - 246 T^{6} - 26 p T^{7} + 48 p^{2} T^{8} + 12 p^{3} T^{9} - 8 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
11 1+2T−23T2−18T3+360T4+26T5−4545T6+26pT7+360p2T8−18p3T9−23p4T10+2p5T11+p6T12 1 + 2 T - 23 T^{2} - 18 T^{3} + 360 T^{4} + 26 T^{5} - 4545 T^{6} + 26 p T^{7} + 360 p^{2} T^{8} - 18 p^{3} T^{9} - 23 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12}
17 1−4T−25T2+116T3+424T4−1320T5−3775T6−1320pT7+424p2T8+116p3T9−25p4T10−4p5T11+p6T12 1 - 4 T - 25 T^{2} + 116 T^{3} + 424 T^{4} - 1320 T^{5} - 3775 T^{6} - 1320 p T^{7} + 424 p^{2} T^{8} + 116 p^{3} T^{9} - 25 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
19 1+4T−42T2−64T3+1670T4+1212T5−34070T6+1212pT7+1670p2T8−64p3T9−42p4T10+4p5T11+p6T12 1 + 4 T - 42 T^{2} - 64 T^{3} + 1670 T^{4} + 1212 T^{5} - 34070 T^{6} + 1212 p T^{7} + 1670 p^{2} T^{8} - 64 p^{3} T^{9} - 42 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
23 1+10T+30T2+52T3+50T4−4230T5−36290T6−4230pT7+50p2T8+52p3T9+30p4T10+10p5T11+p6T12 1 + 10 T + 30 T^{2} + 52 T^{3} + 50 T^{4} - 4230 T^{5} - 36290 T^{6} - 4230 p T^{7} + 50 p^{2} T^{8} + 52 p^{3} T^{9} + 30 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12}
29 (1−24T+272T2−1846T3+272pT4−24p2T5+p3T6)2 ( 1 - 24 T + 272 T^{2} - 1846 T^{3} + 272 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} )^{2}
31 1+4T−58T2−232T3+2126T4+5028T5−55854T6+5028pT7+2126p2T8−232p3T9−58p4T10+4p5T11+p6T12 1 + 4 T - 58 T^{2} - 232 T^{3} + 2126 T^{4} + 5028 T^{5} - 55854 T^{6} + 5028 p T^{7} + 2126 p^{2} T^{8} - 232 p^{3} T^{9} - 58 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
37 1−53T2−248T3+848T4+6572T5+12749T6+6572pT7+848p2T8−248p3T9−53p4T10+p6T12 1 - 53 T^{2} - 248 T^{3} + 848 T^{4} + 6572 T^{5} + 12749 T^{6} + 6572 p T^{7} + 848 p^{2} T^{8} - 248 p^{3} T^{9} - 53 p^{4} T^{10} + p^{6} T^{12}
41 (1+2T+95T2+172T3+95pT4+2p2T5+p3T6)2 ( 1 + 2 T + 95 T^{2} + 172 T^{3} + 95 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
43 (1−10T+58T2−232T3+58pT4−10p2T5+p3T6)2 ( 1 - 10 T + 58 T^{2} - 232 T^{3} + 58 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2}
47 1+8T+2T2+80T3−734T4−16056T5−71134T6−16056pT7−734p2T8+80p3T9+2p4T10+8p5T11+p6T12 1 + 8 T + 2 T^{2} + 80 T^{3} - 734 T^{4} - 16056 T^{5} - 71134 T^{6} - 16056 p T^{7} - 734 p^{2} T^{8} + 80 p^{3} T^{9} + 2 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
53 1+8T−60T2−748T3+1844T4+25200T5+59950T6+25200pT7+1844p2T8−748p3T9−60p4T10+8p5T11+p6T12 1 + 8 T - 60 T^{2} - 748 T^{3} + 1844 T^{4} + 25200 T^{5} + 59950 T^{6} + 25200 p T^{7} + 1844 p^{2} T^{8} - 748 p^{3} T^{9} - 60 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12}
59 1+4T−5T2+516T3+66T4−524T5+375699T6−524pT7+66p2T8+516p3T9−5p4T10+4p5T11+p6T12 1 + 4 T - 5 T^{2} + 516 T^{3} + 66 T^{4} - 524 T^{5} + 375699 T^{6} - 524 p T^{7} + 66 p^{2} T^{8} + 516 p^{3} T^{9} - 5 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
61 (1+2T−57T2+2pT3+p2T4)3 ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{3}
67 1−12T+pT2+340T3−6814T4+21284T5−42709T6+21284pT7−6814p2T8+340p3T9+p5T10−12p5T11+p6T12 1 - 12 T + p T^{2} + 340 T^{3} - 6814 T^{4} + 21284 T^{5} - 42709 T^{6} + 21284 p T^{7} - 6814 p^{2} T^{8} + 340 p^{3} T^{9} + p^{5} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12}
71 (1+6T+191T2+868T3+191pT4+6p2T5+p3T6)2 ( 1 + 6 T + 191 T^{2} + 868 T^{3} + 191 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1+10T−20T2−1172T3−6220T4+20410T5+600530T6+20410pT7−6220p2T8−1172p3T9−20p4T10+10p5T11+p6T12 1 + 10 T - 20 T^{2} - 1172 T^{3} - 6220 T^{4} + 20410 T^{5} + 600530 T^{6} + 20410 p T^{7} - 6220 p^{2} T^{8} - 1172 p^{3} T^{9} - 20 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12}
79 1−14T−46T2+1004T3+8702T4−83502T5−41298T6−83502pT7+8702p2T8+1004p3T9−46p4T10−14p5T11+p6T12 1 - 14 T - 46 T^{2} + 1004 T^{3} + 8702 T^{4} - 83502 T^{5} - 41298 T^{6} - 83502 p T^{7} + 8702 p^{2} T^{8} + 1004 p^{3} T^{9} - 46 p^{4} T^{10} - 14 p^{5} T^{11} + p^{6} T^{12}
83 (1−12T−22T2+1276T3−22pT4−12p2T5+p3T6)2 ( 1 - 12 T - 22 T^{2} + 1276 T^{3} - 22 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2}
89 1−2T−168T2−476T3+14408T4+56742T5−1250366T6+56742pT7+14408p2T8−476p3T9−168p4T10−2p5T11+p6T12 1 - 2 T - 168 T^{2} - 476 T^{3} + 14408 T^{4} + 56742 T^{5} - 1250366 T^{6} + 56742 p T^{7} + 14408 p^{2} T^{8} - 476 p^{3} T^{9} - 168 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12}
97 (1−10T+320T2−1962T3+320pT4−10p2T5+p3T6)2 ( 1 - 10 T + 320 T^{2} - 1962 T^{3} + 320 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2}
show more
show less
   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

−5.78675076298302276214829733258, −5.46355984273638963438637246969, −5.31206813958636241939650218628, −5.18039264175491817948144228020, −4.84146415692001326045193405397, −4.75908726347504027279103993852, −4.70573741139571697393734194802, −4.57077906792325200306757563257, −4.53804492631684499752186101669, −4.21252836670046350555412295825, −4.15657259153233308153630103206, −3.44331498491179379180998787756, −3.30609991253543062908775183409, −3.29570918993894927309806988445, −2.98731211646972824621625050261, −2.84483254243896179756225455504, −2.70773884382702482345612865720, −2.22097348516636377598838239691, −2.18656468621651464154888870987, −2.13434543047120394955402936252, −1.75351923697494255766619066191, −1.17963907157478417566298123936, −0.983812211928118496442745611831, −0.833195135585768240361321609726, −0.56694498645074791712338353965, 0.56694498645074791712338353965, 0.833195135585768240361321609726, 0.983812211928118496442745611831, 1.17963907157478417566298123936, 1.75351923697494255766619066191, 2.13434543047120394955402936252, 2.18656468621651464154888870987, 2.22097348516636377598838239691, 2.70773884382702482345612865720, 2.84483254243896179756225455504, 2.98731211646972824621625050261, 3.29570918993894927309806988445, 3.30609991253543062908775183409, 3.44331498491179379180998787756, 4.15657259153233308153630103206, 4.21252836670046350555412295825, 4.53804492631684499752186101669, 4.57077906792325200306757563257, 4.70573741139571697393734194802, 4.75908726347504027279103993852, 4.84146415692001326045193405397, 5.18039264175491817948144228020, 5.31206813958636241939650218628, 5.46355984273638963438637246969, 5.78675076298302276214829733258

Graph of the ZZ-function along the critical line

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