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Properties

Label 12-1134e6-1.1-c1e6-0-3
Degree 1212
Conductor 2.127×10182.127\times 10^{18}
Sign 11
Analytic cond. 551240.551240.
Root an. cond. 3.009153.00915
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 + 3·4-s − 5·5-s − 2·7-s + 2·8-s + 15·10-s − 11-s + 4·13-s + 6·14-s − 9·16-s − 4·17-s − 3·19-s − 15·20-s + 3·22-s − 7·23-s + 19·25-s − 12·26-s − 6·28-s + 10·29-s − 14·31-s + 9·32-s + 12·34-s + 10·35-s − 9·37-s + 9·38-s − 10·40-s + 24·41-s + ⋯
L(s)  = 1  − 2.12·2-s + 3/2·4-s − 2.23·5-s − 0.755·7-s + 0.707·8-s + 4.74·10-s − 0.301·11-s + 1.10·13-s + 1.60·14-s − 9/4·16-s − 0.970·17-s − 0.688·19-s − 3.35·20-s + 0.639·22-s − 1.45·23-s + 19/5·25-s − 2.35·26-s − 1.13·28-s + 1.85·29-s − 2.51·31-s + 1.59·32-s + 2.05·34-s + 1.69·35-s − 1.47·37-s + 1.45·38-s − 1.58·40-s + 3.74·41-s + ⋯

Functional equation

Λ(s)=((26⋅324⋅76)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((26⋅324⋅76)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 26⋅324⋅762^{6} \cdot 3^{24} \cdot 7^{6}
Sign: 11
Analytic conductor: 551240.551240.
Root analytic conductor: 3.009153.00915
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 26⋅324⋅76, ( :[1/2]6), 1)(12,\ 2^{6} \cdot 3^{24} \cdot 7^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.28917631300.2891763130
L(12)L(\frac12) ≈\approx 0.28917631300.2891763130
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+T2)3 ( 1 + T + T^{2} )^{3}
3 1 1
7 1+2T−4T2−31T3−4pT4+2p2T5+p3T6 1 + 2 T - 4 T^{2} - 31 T^{3} - 4 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
good5 1+pT+6T2+T3+31T4+68T5+29T6+68pT7+31p2T8+p3T9+6p4T10+p6T11+p6T12 1 + p T + 6 T^{2} + T^{3} + 31 T^{4} + 68 T^{5} + 29 T^{6} + 68 p T^{7} + 31 p^{2} T^{8} + p^{3} T^{9} + 6 p^{4} T^{10} + p^{6} T^{11} + p^{6} T^{12}
11 1+T−6T2−103T3−83T4+32pT5+457pT6+32p2T7−83p2T8−103p3T9−6p4T10+p5T11+p6T12 1 + T - 6 T^{2} - 103 T^{3} - 83 T^{4} + 32 p T^{5} + 457 p T^{6} + 32 p^{2} T^{7} - 83 p^{2} T^{8} - 103 p^{3} T^{9} - 6 p^{4} T^{10} + p^{5} T^{11} + p^{6} T^{12}
13 (1−2T+36T2−49T3+36pT4−2p2T5+p3T6)2 ( 1 - 2 T + 36 T^{2} - 49 T^{3} + 36 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2}
17 1+4T+9T2+92T3+58T4−20T5+5393T6−20pT7+58p2T8+92p3T9+9p4T10+4p5T11+p6T12 1 + 4 T + 9 T^{2} + 92 T^{3} + 58 T^{4} - 20 T^{5} + 5393 T^{6} - 20 p T^{7} + 58 p^{2} T^{8} + 92 p^{3} T^{9} + 9 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12}
19 1+3T−42T2−61T3+69pT4+726T5−27501T6+726pT7+69p3T8−61p3T9−42p4T10+3p5T11+p6T12 1 + 3 T - 42 T^{2} - 61 T^{3} + 69 p T^{4} + 726 T^{5} - 27501 T^{6} + 726 p T^{7} + 69 p^{3} T^{8} - 61 p^{3} T^{9} - 42 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12}
23 1+7T−24T2−127T3+1417T4+3484T5−22393T6+3484pT7+1417p2T8−127p3T9−24p4T10+7p5T11+p6T12 1 + 7 T - 24 T^{2} - 127 T^{3} + 1417 T^{4} + 3484 T^{5} - 22393 T^{6} + 3484 p T^{7} + 1417 p^{2} T^{8} - 127 p^{3} T^{9} - 24 p^{4} T^{10} + 7 p^{5} T^{11} + p^{6} T^{12}
29 (1−5T+55T2−323T3+55pT4−5p2T5+p3T6)2 ( 1 - 5 T + 55 T^{2} - 323 T^{3} + 55 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} )^{2}
31 1+14T+58T2+250T3+2992T4+9728T5−11857T6+9728pT7+2992p2T8+250p3T9+58p4T10+14p5T11+p6T12 1 + 14 T + 58 T^{2} + 250 T^{3} + 2992 T^{4} + 9728 T^{5} - 11857 T^{6} + 9728 p T^{7} + 2992 p^{2} T^{8} + 250 p^{3} T^{9} + 58 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12}
37 1+9T−21T2−268T3+1293T4+4875T5−42882T6+4875pT7+1293p2T8−268p3T9−21p4T10+9p5T11+p6T12 1 + 9 T - 21 T^{2} - 268 T^{3} + 1293 T^{4} + 4875 T^{5} - 42882 T^{6} + 4875 p T^{7} + 1293 p^{2} T^{8} - 268 p^{3} T^{9} - 21 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12}
41 (1−12T+162T2−1011T3+162pT4−12p2T5+p3T6)2 ( 1 - 12 T + 162 T^{2} - 1011 T^{3} + 162 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} )^{2}
43 (1+18T+210T2+1549T3+210pT4+18p2T5+p3T6)2 ( 1 + 18 T + 210 T^{2} + 1549 T^{3} + 210 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} )^{2}
47 1−3T−108T2+267T3+7263T4−9786T5−360137T6−9786pT7+7263p2T8+267p3T9−108p4T10−3p5T11+p6T12 1 - 3 T - 108 T^{2} + 267 T^{3} + 7263 T^{4} - 9786 T^{5} - 360137 T^{6} - 9786 p T^{7} + 7263 p^{2} T^{8} + 267 p^{3} T^{9} - 108 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12}
53 1−9T−36T2+873T3−1179T4−26334T5+272077T6−26334pT7−1179p2T8+873p3T9−36p4T10−9p5T11+p6T12 1 - 9 T - 36 T^{2} + 873 T^{3} - 1179 T^{4} - 26334 T^{5} + 272077 T^{6} - 26334 p T^{7} - 1179 p^{2} T^{8} + 873 p^{3} T^{9} - 36 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12}
59 1−4T−60T2+994T3−1304T4−464pT5+7381pT6−464p2T7−1304p2T8+994p3T9−60p4T10−4p5T11+p6T12 1 - 4 T - 60 T^{2} + 994 T^{3} - 1304 T^{4} - 464 p T^{5} + 7381 p T^{6} - 464 p^{2} T^{7} - 1304 p^{2} T^{8} + 994 p^{3} T^{9} - 60 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
61 1−4T−32T2−650T3+292T4+19532T5+306323T6+19532pT7+292p2T8−650p3T9−32p4T10−4p5T11+p6T12 1 - 4 T - 32 T^{2} - 650 T^{3} + 292 T^{4} + 19532 T^{5} + 306323 T^{6} + 19532 p T^{7} + 292 p^{2} T^{8} - 650 p^{3} T^{9} - 32 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12}
67 1−5T−118T2+327T3+8263T4−1138T5−609341T6−1138pT7+8263p2T8+327p3T9−118p4T10−5p5T11+p6T12 1 - 5 T - 118 T^{2} + 327 T^{3} + 8263 T^{4} - 1138 T^{5} - 609341 T^{6} - 1138 p T^{7} + 8263 p^{2} T^{8} + 327 p^{3} T^{9} - 118 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12}
71 (1−7T+163T2−895T3+163pT4−7p2T5+p3T6)2 ( 1 - 7 T + 163 T^{2} - 895 T^{3} + 163 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1+25T+254T2+2073T3+20533T4+115046T5+366817T6+115046pT7+20533p2T8+2073p3T9+254p4T10+25p5T11+p6T12 1 + 25 T + 254 T^{2} + 2073 T^{3} + 20533 T^{4} + 115046 T^{5} + 366817 T^{6} + 115046 p T^{7} + 20533 p^{2} T^{8} + 2073 p^{3} T^{9} + 254 p^{4} T^{10} + 25 p^{5} T^{11} + p^{6} T^{12}
79 1−7T−44T2+19T3−1043T4+28016T5+109223T6+28016pT7−1043p2T8+19p3T9−44p4T10−7p5T11+p6T12 1 - 7 T - 44 T^{2} + 19 T^{3} - 1043 T^{4} + 28016 T^{5} + 109223 T^{6} + 28016 p T^{7} - 1043 p^{2} T^{8} + 19 p^{3} T^{9} - 44 p^{4} T^{10} - 7 p^{5} T^{11} + p^{6} T^{12}
83 (1+8T+244T2+1235T3+244pT4+8p2T5+p3T6)2 ( 1 + 8 T + 244 T^{2} + 1235 T^{3} + 244 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
89 1+9T−180T2−729T3+31041T4+54846T5−2925911T6+54846pT7+31041p2T8−729p3T9−180p4T10+9p5T11+p6T12 1 + 9 T - 180 T^{2} - 729 T^{3} + 31041 T^{4} + 54846 T^{5} - 2925911 T^{6} + 54846 p T^{7} + 31041 p^{2} T^{8} - 729 p^{3} T^{9} - 180 p^{4} T^{10} + 9 p^{5} T^{11} + p^{6} T^{12}
97 (1−28T+527T2−5968T3+527pT4−28p2T5+p3T6)2 ( 1 - 28 T + 527 T^{2} - 5968 T^{3} + 527 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} )^{2}
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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

−5.19519306690671494788902375234, −4.96299467513367235472222762136, −4.79829197923494158258656653615, −4.66488255065249405452164573364, −4.58955964254241696566197830586, −4.19448576483144773271991219079, −4.01085434130402708205454344513, −4.00584817743247724348132592096, −3.96751819356929634599813853955, −3.87694579987052817482251669890, −3.43342364269552035684143509852, −3.37364665683261707268066279128, −3.08866653232063457091120657261, −3.07625436060763517994838722589, −2.88533625539755533063736273890, −2.35686388516774258797555001776, −2.27646545032197456526097968464, −2.15340400706614926263789008071, −1.95596056654158531996416925831, −1.39629836555097936674248722684, −1.33172345444547014894403688882, −1.25383323382634493220320263644, −0.49050841873831015182161904990, −0.41508531589434567343333966809, −0.39272297397854271238514644703, 0.39272297397854271238514644703, 0.41508531589434567343333966809, 0.49050841873831015182161904990, 1.25383323382634493220320263644, 1.33172345444547014894403688882, 1.39629836555097936674248722684, 1.95596056654158531996416925831, 2.15340400706614926263789008071, 2.27646545032197456526097968464, 2.35686388516774258797555001776, 2.88533625539755533063736273890, 3.07625436060763517994838722589, 3.08866653232063457091120657261, 3.37364665683261707268066279128, 3.43342364269552035684143509852, 3.87694579987052817482251669890, 3.96751819356929634599813853955, 4.00584817743247724348132592096, 4.01085434130402708205454344513, 4.19448576483144773271991219079, 4.58955964254241696566197830586, 4.66488255065249405452164573364, 4.79829197923494158258656653615, 4.96299467513367235472222762136, 5.19519306690671494788902375234

Graph of the ZZ-function along the critical line

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