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Properties

Label 24-60e12-1.1-c1e12-0-0
Degree 2424
Conductor 2.177×10212.177\times 10^{21}
Sign 11
Analytic cond. 0.0001462640.000146264
Root an. cond. 0.6921720.692172
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  − 4·8-s − 4·13-s + 3·16-s − 20·17-s − 10·25-s + 4·32-s + 4·37-s + 16·41-s + 4·53-s − 32·61-s + 8·64-s + 44·73-s − 3·81-s − 20·97-s − 40·101-s + 16·104-s − 52·113-s + 60·121-s − 16·125-s + 127-s − 16·128-s + 131-s + 80·136-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.41·8-s − 1.10·13-s + 3/4·16-s − 4.85·17-s − 2·25-s + 0.707·32-s + 0.657·37-s + 2.49·41-s + 0.549·53-s − 4.09·61-s + 64-s + 5.14·73-s − 1/3·81-s − 2.03·97-s − 3.98·101-s + 1.56·104-s − 4.89·113-s + 5.45·121-s − 1.43·125-s + 0.0887·127-s − 1.41·128-s + 0.0873·131-s + 6.85·136-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

Λ(s)=((224⋅312⋅512)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((224⋅312⋅512)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 224⋅312⋅5122^{24} \cdot 3^{12} \cdot 5^{12}
Sign: 11
Analytic conductor: 0.0001462640.000146264
Root analytic conductor: 0.6921720.692172
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 224⋅312⋅512, ( :[1/2]12), 1)(24,\ 2^{24} \cdot 3^{12} \cdot 5^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.17098213600.1709821360
L(12)L(\frac12) ≈\approx 0.17098213600.1709821360
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+p2T3−3T4−p2T5+p3T6−p3T7−3p2T8+p5T9+p6T12 1 + p^{2} T^{3} - 3 T^{4} - p^{2} T^{5} + p^{3} T^{6} - p^{3} T^{7} - 3 p^{2} T^{8} + p^{5} T^{9} + p^{6} T^{12}
3 (1+T4)3 ( 1 + T^{4} )^{3}
5 (1+pT2+8T3+p2T4+p3T6)2 ( 1 + p T^{2} + 8 T^{3} + p^{2} T^{4} + p^{3} T^{6} )^{2}
good7 1−66T4+671T8+41476T12+671p4T16−66p8T20+p12T24 1 - 66 T^{4} + 671 T^{8} + 41476 T^{12} + 671 p^{4} T^{16} - 66 p^{8} T^{20} + p^{12} T^{24}
11 (1−30T2+491T4−6076T6+491p2T8−30p4T10+p6T12)2 ( 1 - 30 T^{2} + 491 T^{4} - 6076 T^{6} + 491 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} )^{2}
13 (1+2T+2T2−6T3+27T4+580T5+1124T6+580pT7+27p2T8−6p3T9+2p4T10+2p5T11+p6T12)2 ( 1 + 2 T + 2 T^{2} - 6 T^{3} + 27 T^{4} + 580 T^{5} + 1124 T^{6} + 580 p T^{7} + 27 p^{2} T^{8} - 6 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2}
17 (1+10T+50T2+250T3+1155T4+4260T5+16100T6+4260pT7+1155p2T8+250p3T9+50p4T10+10p5T11+p6T12)2 ( 1 + 10 T + 50 T^{2} + 250 T^{3} + 1155 T^{4} + 4260 T^{5} + 16100 T^{6} + 4260 p T^{7} + 1155 p^{2} T^{8} + 250 p^{3} T^{9} + 50 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2}
19 (1+74T2+2775T4+65228T6+2775p2T8+74p4T10+p6T12)2 ( 1 + 74 T^{2} + 2775 T^{4} + 65228 T^{6} + 2775 p^{2} T^{8} + 74 p^{4} T^{10} + p^{6} T^{12} )^{2}
23 1+1190T4+607727T8+223349076T12+607727p4T16+1190p8T20+p12T24 1 + 1190 T^{4} + 607727 T^{8} + 223349076 T^{12} + 607727 p^{4} T^{16} + 1190 p^{8} T^{20} + p^{12} T^{24}
29 (1−154T2+10395T4−392596T6+10395p2T8−154p4T10+p6T12)2 ( 1 - 154 T^{2} + 10395 T^{4} - 392596 T^{6} + 10395 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} )^{2}
31 (1−74T2+4175T4−143436T6+4175p2T8−74p4T10+p6T12)2 ( 1 - 74 T^{2} + 4175 T^{4} - 143436 T^{6} + 4175 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} )^{2}
37 (1−10T+93T2−472T3+93pT4−10p2T5+p3T6)2(1+8T−11T2−424T3−11pT4+8p2T5+p3T6)2 ( 1 - 10 T + 93 T^{2} - 472 T^{3} + 93 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2}( 1 + 8 T - 11 T^{2} - 424 T^{3} - 11 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2}
41 (1−4T+103T2−264T3+103pT4−4p2T5+p3T6)4 ( 1 - 4 T + 103 T^{2} - 264 T^{3} + 103 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{4}
43 1−6826T4+23542879T8−52748850124T12+23542879p4T16−6826p8T20+p12T24 1 - 6826 T^{4} + 23542879 T^{8} - 52748850124 T^{12} + 23542879 p^{4} T^{16} - 6826 p^{8} T^{20} + p^{12} T^{24}
47 1+1606T4+1555663T8−11534742380T12+1555663p4T16+1606p8T20+p12T24 1 + 1606 T^{4} + 1555663 T^{8} - 11534742380 T^{12} + 1555663 p^{4} T^{16} + 1606 p^{8} T^{20} + p^{12} T^{24}
53 (1−2T+2T2−90T3+5291T4−7252T5+7972T6−7252pT7+5291p2T8−90p3T9+2p4T10−2p5T11+p6T12)2 ( 1 - 2 T + 2 T^{2} - 90 T^{3} + 5291 T^{4} - 7252 T^{5} + 7972 T^{6} - 7252 p T^{7} + 5291 p^{2} T^{8} - 90 p^{3} T^{9} + 2 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2}
59 (1+254T2+30475T4+2237948T6+30475p2T8+254p4T10+p6T12)2 ( 1 + 254 T^{2} + 30475 T^{4} + 2237948 T^{6} + 30475 p^{2} T^{8} + 254 p^{4} T^{10} + p^{6} T^{12} )^{2}
61 (1+8T+83T2+1152T3+83pT4+8p2T5+p3T6)4 ( 1 + 8 T + 83 T^{2} + 1152 T^{3} + 83 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4}
67 1−6346T4+60212671T8−234156159244T12+60212671p4T16−6346p8T20+p12T24 1 - 6346 T^{4} + 60212671 T^{8} - 234156159244 T^{12} + 60212671 p^{4} T^{16} - 6346 p^{8} T^{20} + p^{12} T^{24}
71 (1−170T2+18527T4−1427916T6+18527p2T8−170p4T10+p6T12)2 ( 1 - 170 T^{2} + 18527 T^{4} - 1427916 T^{6} + 18527 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} )^{2}
73 (1−22T+242T2−2246T3+20271T4−179604T5+1567964T6−179604pT7+20271p2T8−2246p3T9+242p4T10−22p5T11+p6T12)2 ( 1 - 22 T + 242 T^{2} - 2246 T^{3} + 20271 T^{4} - 179604 T^{5} + 1567964 T^{6} - 179604 p T^{7} + 20271 p^{2} T^{8} - 2246 p^{3} T^{9} + 242 p^{4} T^{10} - 22 p^{5} T^{11} + p^{6} T^{12} )^{2}
79 (1+170T2+10415T4+507660T6+10415p2T8+170p4T10+p6T12)2 ( 1 + 170 T^{2} + 10415 T^{4} + 507660 T^{6} + 10415 p^{2} T^{8} + 170 p^{4} T^{10} + p^{6} T^{12} )^{2}
83 1+7542T4−16589569T8−295904039564T12−16589569p4T16+7542p8T20+p12T24 1 + 7542 T^{4} - 16589569 T^{8} - 295904039564 T^{12} - 16589569 p^{4} T^{16} + 7542 p^{8} T^{20} + p^{12} T^{24}
89 (1−462T2+94223T4−10861604T6+94223p2T8−462p4T10+p6T12)2 ( 1 - 462 T^{2} + 94223 T^{4} - 10861604 T^{6} + 94223 p^{2} T^{8} - 462 p^{4} T^{10} + p^{6} T^{12} )^{2}
97 (1+10T+50T2−1078T3−9153T4+45804T5+1496732T6+45804pT7−9153p2T8−1078p3T9+50p4T10+10p5T11+p6T12)2 ( 1 + 10 T + 50 T^{2} - 1078 T^{3} - 9153 T^{4} + 45804 T^{5} + 1496732 T^{6} + 45804 p T^{7} - 9153 p^{2} T^{8} - 1078 p^{3} T^{9} + 50 p^{4} T^{10} + 10 p^{5} T^{11} + p^{6} T^{12} )^{2}
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   L(s)=∏p ∏j=124(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.80193236186153991877192141527, −5.64597234276793293522754012365, −5.57950822027377763077666186519, −5.43689528568042674304442276403, −5.35152973487716363715185917574, −5.12045987168770241831574206311, −5.00239101024269964312071468942, −4.86935030706652868948885848305, −4.53011808042202940201052835960, −4.42259064662844709064730415666, −4.35002774704167954760368500884, −4.22186681257837288851574594309, −4.10845597245463350886600356084, −4.10820850581918588614915726140, −3.96505536266266673223687962790, −3.38477227989812797661300606691, −3.37130257409424508583511936349, −3.23476554326180866955618992189, −2.99794960363558667981740078565, −2.62726573415258421154532584130, −2.45183576347053094532963864216, −2.42227100475349896467561560502, −2.35091786896788376344780558035, −1.87502238686416301347343536366, −1.63912336428237542663196072666, 1.63912336428237542663196072666, 1.87502238686416301347343536366, 2.35091786896788376344780558035, 2.42227100475349896467561560502, 2.45183576347053094532963864216, 2.62726573415258421154532584130, 2.99794960363558667981740078565, 3.23476554326180866955618992189, 3.37130257409424508583511936349, 3.38477227989812797661300606691, 3.96505536266266673223687962790, 4.10820850581918588614915726140, 4.10845597245463350886600356084, 4.22186681257837288851574594309, 4.35002774704167954760368500884, 4.42259064662844709064730415666, 4.53011808042202940201052835960, 4.86935030706652868948885848305, 5.00239101024269964312071468942, 5.12045987168770241831574206311, 5.35152973487716363715185917574, 5.43689528568042674304442276403, 5.57950822027377763077666186519, 5.64597234276793293522754012365, 5.80193236186153991877192141527

Graph of the ZZ-function along the critical line

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