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Properties

Label 12-2400e6-1.1-c1e6-0-2
Degree 1212
Conductor 1.911×10201.911\times 10^{20}
Sign 11
Analytic cond. 4.95370×1074.95370\times 10^{7}
Root an. cond. 4.377684.37768
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·3-s + 21·9-s + 56·27-s + 12·31-s − 8·37-s − 20·41-s − 8·43-s + 6·49-s − 12·53-s + 24·67-s + 8·71-s + 36·79-s + 126·81-s + 32·83-s + 28·89-s + 72·93-s + 40·107-s − 48·111-s + 2·121-s − 120·123-s + 127-s − 48·129-s + 131-s + 137-s + 139-s + 36·147-s + 149-s + ⋯
L(s)  = 1  + 3.46·3-s + 7·9-s + 10.7·27-s + 2.15·31-s − 1.31·37-s − 3.12·41-s − 1.21·43-s + 6/7·49-s − 1.64·53-s + 2.93·67-s + 0.949·71-s + 4.05·79-s + 14·81-s + 3.51·83-s + 2.96·89-s + 7.46·93-s + 3.86·107-s − 4.55·111-s + 2/11·121-s − 10.8·123-s + 0.0887·127-s − 4.22·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.96·147-s + 0.0819·149-s + ⋯

Functional equation

Λ(s)=((230⋅36⋅512)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((230⋅36⋅512)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 230⋅36⋅5122^{30} \cdot 3^{6} \cdot 5^{12}
Sign: 11
Analytic conductor: 4.95370×1074.95370\times 10^{7}
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 230⋅36⋅512, ( :[1/2]6), 1)(12,\ 2^{30} \cdot 3^{6} \cdot 5^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 40.2701110140.27011101
L(12)L(\frac12) ≈\approx 40.2701110140.27011101
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 1
3 (1−T)6 ( 1 - T )^{6}
5 1 1
good7 1−6T2+47T4−500T6+47p2T8−6p4T10+p6T12 1 - 6 T^{2} + 47 T^{4} - 500 T^{6} + 47 p^{2} T^{8} - 6 p^{4} T^{10} + p^{6} T^{12}
11 1−2T2+87T4+4T6+87p2T8−2p4T10+p6T12 1 - 2 T^{2} + 87 T^{4} + 4 T^{6} + 87 p^{2} T^{8} - 2 p^{4} T^{10} + p^{6} T^{12}
13 (1+11T2−16T3+11pT4+p3T6)2 ( 1 + 11 T^{2} - 16 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{2}
17 1−2pT2+351T4−1084T6+351p2T8−2p5T10+p6T12 1 - 2 p T^{2} + 351 T^{4} - 1084 T^{6} + 351 p^{2} T^{8} - 2 p^{5} T^{10} + p^{6} T^{12}
19 1−74T2+2647T4−60620T6+2647p2T8−74p4T10+p6T12 1 - 74 T^{2} + 2647 T^{4} - 60620 T^{6} + 2647 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12}
23 1−98T2+4527T4−128636T6+4527p2T8−98p4T10+p6T12 1 - 98 T^{2} + 4527 T^{4} - 128636 T^{6} + 4527 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12}
29 (1−54T2+p2T4)3 ( 1 - 54 T^{2} + p^{2} T^{4} )^{3}
31 (1−6T+77T2−308T3+77pT4−6p2T5+p3T6)2 ( 1 - 6 T + 77 T^{2} - 308 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2}
37 (1+4T+51T2+40T3+51pT4+4p2T5+p3T6)2 ( 1 + 4 T + 51 T^{2} + 40 T^{3} + 51 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2}
41 (1+10T+87T2+588T3+87pT4+10p2T5+p3T6)2 ( 1 + 10 T + 87 T^{2} + 588 T^{3} + 87 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2}
43 (1+4T+65T2+216T3+65pT4+4p2T5+p3T6)2 ( 1 + 4 T + 65 T^{2} + 216 T^{3} + 65 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2}
47 1−82T2+7967T4−348252T6+7967p2T8−82p4T10+p6T12 1 - 82 T^{2} + 7967 T^{4} - 348252 T^{6} + 7967 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12}
53 (1+2T+pT2)6 ( 1 + 2 T + p T^{2} )^{6}
59 1−274T2+33911T4−2503644T6+33911p2T8−274p4T10+p6T12 1 - 274 T^{2} + 33911 T^{4} - 2503644 T^{6} + 33911 p^{2} T^{8} - 274 p^{4} T^{10} + p^{6} T^{12}
61 1−110T2+10759T4−685796T6+10759p2T8−110p4T10+p6T12 1 - 110 T^{2} + 10759 T^{4} - 685796 T^{6} + 10759 p^{2} T^{8} - 110 p^{4} T^{10} + p^{6} T^{12}
67 (1−4T+pT2)6 ( 1 - 4 T + p T^{2} )^{6}
71 (1−4T+101T2−632T3+101pT4−4p2T5+p3T6)2 ( 1 - 4 T + 101 T^{2} - 632 T^{3} + 101 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 (1−16T+pT2)3(1+16T+pT2)3 ( 1 - 16 T + p T^{2} )^{3}( 1 + 16 T + p T^{2} )^{3}
79 (1−18T+317T2−2908T3+317pT4−18p2T5+p3T6)2 ( 1 - 18 T + 317 T^{2} - 2908 T^{3} + 317 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} )^{2}
83 (1−16T+265T2−2400T3+265pT4−16p2T5+p3T6)2 ( 1 - 16 T + 265 T^{2} - 2400 T^{3} + 265 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} )^{2}
89 (1−14T+263T2−2308T3+263pT4−14p2T5+p3T6)2 ( 1 - 14 T + 263 T^{2} - 2308 T^{3} + 263 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{2}
97 1−250T2+24143T4−1697004T6+24143p2T8−250p4T10+p6T12 1 - 250 T^{2} + 24143 T^{4} - 1697004 T^{6} + 24143 p^{2} T^{8} - 250 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

−4.75548612646811221649308172778, −4.56414167715133082915611853668, −4.23828884846416023973063054672, −4.00636078270904823089089889224, −3.95006113287529293088429752837, −3.68607614862327999040566169900, −3.58914634324196703489370340258, −3.58259352866861253422616305858, −3.39856518645524919302969834009, −3.32996823022288471911372854979, −3.29770649476703901681287696962, −2.96919658027631263968802931052, −2.78850816464886513641144076088, −2.49583725700872458240328398898, −2.47695400933385392709423147985, −2.17524481150253556806387924748, −2.16922650646852252975210900009, −2.05236991245424410707104141295, −1.86004755616638252934571693435, −1.64652685770277551560491194768, −1.36528714491591577329387285995, −1.20203360544240795621066385997, −0.75956929801200665649749317828, −0.75908279633423591072970556629, −0.37674848081668442111708560378, 0.37674848081668442111708560378, 0.75908279633423591072970556629, 0.75956929801200665649749317828, 1.20203360544240795621066385997, 1.36528714491591577329387285995, 1.64652685770277551560491194768, 1.86004755616638252934571693435, 2.05236991245424410707104141295, 2.16922650646852252975210900009, 2.17524481150253556806387924748, 2.47695400933385392709423147985, 2.49583725700872458240328398898, 2.78850816464886513641144076088, 2.96919658027631263968802931052, 3.29770649476703901681287696962, 3.32996823022288471911372854979, 3.39856518645524919302969834009, 3.58259352866861253422616305858, 3.58914634324196703489370340258, 3.68607614862327999040566169900, 3.95006113287529293088429752837, 4.00636078270904823089089889224, 4.23828884846416023973063054672, 4.56414167715133082915611853668, 4.75548612646811221649308172778

Graph of the ZZ-function along the critical line

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