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Properties

Label 24-3e72-1.1-c1e12-0-6
Degree 2424
Conductor 2.253×10342.253\times 10^{34}
Sign 11
Analytic cond. 1.51375×1091.51375\times 10^{9}
Root an. cond. 2.412692.41269
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  + 6·19-s − 48·37-s + 16·64-s − 66·73-s − 12·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯
L(s)  = 1  + 1.37·19-s − 7.89·37-s + 2·64-s − 7.72·73-s − 1.14·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + ⋯

Functional equation

Λ(s)=((372)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((372)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{72}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 3723^{72}
Sign: 11
Analytic conductor: 1.51375×1091.51375\times 10^{9}
Root analytic conductor: 2.412692.41269
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 372, ( :[1/2]12), 1)(24,\ 3^{72} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.0567132561.056713256
L(12)L(\frac12) ≈\approx 1.0567132561.056713256
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)
bad3 1 1
good2 (1−p3T6+p6T12)2 ( 1 - p^{3} T^{6} + p^{6} T^{12} )^{2}
5 1+236T6+40071T12+236p6T18+p12T24 1 + 236 T^{6} + 40071 T^{12} + 236 p^{6} T^{18} + p^{12} T^{24}
7 (1−34T3+813T6−34p3T9+p6T12)2 ( 1 - 34 T^{3} + 813 T^{6} - 34 p^{3} T^{9} + p^{6} T^{12} )^{2}
11 1+1712T6+1159383T12+1712p6T18+p12T24 1 + 1712 T^{6} + 1159383 T^{12} + 1712 p^{6} T^{18} + p^{12} T^{24}
13 (1+38T3−753T6+38p3T9+p6T12)2 ( 1 + 38 T^{3} - 753 T^{6} + 38 p^{3} T^{9} + p^{6} T^{12} )^{2}
17 (1+20T2+111T4+20p2T6+p4T8)3 ( 1 + 20 T^{2} + 111 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{3}
19 (1−8T+pT2)6(1+7T+pT2)6 ( 1 - 8 T + p T^{2} )^{6}( 1 + 7 T + p T^{2} )^{6}
23 1−520T6−147765489T12−520p6T18+p12T24 1 - 520 T^{6} - 147765489 T^{12} - 520 p^{6} T^{18} + p^{12} T^{24}
29 1+46478T6+1565381163T12+46478p6T18+p12T24 1 + 46478 T^{6} + 1565381163 T^{12} + 46478 p^{6} T^{18} + p^{12} T^{24}
31 (1+92T3−21327T6+92p3T9+p6T12)2 ( 1 + 92 T^{3} - 21327 T^{6} + 92 p^{3} T^{9} + p^{6} T^{12} )^{2}
37 (1+8T+27T2+8pT3+p2T4)6 ( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{6}
41 1+97382T6+4733149683T12+97382p6T18+p12T24 1 + 97382 T^{6} + 4733149683 T^{12} + 97382 p^{6} T^{18} + p^{12} T^{24}
43 (1−88T3−71763T6−88p3T9+p6T12)2 ( 1 - 88 T^{3} - 71763 T^{6} - 88 p^{3} T^{9} + p^{6} T^{12} )^{2}
47 1−13246T6−10603758813T12−13246p6T18+p12T24 1 - 13246 T^{6} - 10603758813 T^{12} - 13246 p^{6} T^{18} + p^{12} T^{24}
53 (1+52T2+p2T4)6 ( 1 + 52 T^{2} + p^{2} T^{4} )^{6}
59 1−235312T6+13191203703T12−235312p6T18+p12T24 1 - 235312 T^{6} + 13191203703 T^{12} - 235312 p^{6} T^{18} + p^{12} T^{24}
61 (1−790T3+397119T6−790p3T9+p6T12)2 ( 1 - 790 T^{3} + 397119 T^{6} - 790 p^{3} T^{9} + p^{6} T^{12} )^{2}
67 (1+1064T3+831333T6+1064p3T9+p6T12)2 ( 1 + 1064 T^{3} + 831333 T^{6} + 1064 p^{3} T^{9} + p^{6} T^{12} )^{2}
71 (1−88T2+2703T4−88p2T6+p4T8)3 ( 1 - 88 T^{2} + 2703 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} )^{3}
73 (1+11T+48T2+11pT3+p2T4)6 ( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{6}
79 (1+1316T3+1238817T6+1316p3T9+p6T12)2 ( 1 + 1316 T^{3} + 1238817 T^{6} + 1316 p^{3} T^{9} + p^{6} T^{12} )^{2}
83 1+326576T6−220288489593T12+326576p6T18+p12T24 1 + 326576 T^{6} - 220288489593 T^{12} + 326576 p^{6} T^{18} + p^{12} T^{24}
89 (1−pT2+p2T4)6 ( 1 - p T^{2} + p^{2} T^{4} )^{6}
97 (1+1694T3+1956963T6+1694p3T9+p6T12)2 ( 1 + 1694 T^{3} + 1956963 T^{6} + 1694 p^{3} T^{9} + 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

−3.38820118557286355417146479891, −3.34856335420024081147512409483, −3.30087905215984719612479392837, −3.27034683558264249916914133659, −2.91097444701345467779875168704, −2.87677162167449744720379689589, −2.70296862158360008463576048151, −2.61337457262061862131272820235, −2.55284369076679863162709418970, −2.54071501279500209673692587599, −2.46559426982897407613324545775, −2.12438216014932638184035308650, −2.06588318468725458854096480770, −1.91186421346557494617224878107, −1.80280562456634389175073435232, −1.67425066860703962612759841841, −1.58199364083437925803951477050, −1.43767813133883439299441713143, −1.43262748222905769819619402922, −1.32740546626422994784308053203, −1.09503061048420658879406261007, −0.828688624491273752995699474596, −0.47832071229389434722840464672, −0.39185498606565115252436813155, −0.13300248356930204438894034282, 0.13300248356930204438894034282, 0.39185498606565115252436813155, 0.47832071229389434722840464672, 0.828688624491273752995699474596, 1.09503061048420658879406261007, 1.32740546626422994784308053203, 1.43262748222905769819619402922, 1.43767813133883439299441713143, 1.58199364083437925803951477050, 1.67425066860703962612759841841, 1.80280562456634389175073435232, 1.91186421346557494617224878107, 2.06588318468725458854096480770, 2.12438216014932638184035308650, 2.46559426982897407613324545775, 2.54071501279500209673692587599, 2.55284369076679863162709418970, 2.61337457262061862131272820235, 2.70296862158360008463576048151, 2.87677162167449744720379689589, 2.91097444701345467779875168704, 3.27034683558264249916914133659, 3.30087905215984719612479392837, 3.34856335420024081147512409483, 3.38820118557286355417146479891

Graph of the ZZ-function along the critical line

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