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Properties

Label 24-675e12-1.1-c1e12-0-1
Degree 2424
Conductor 8.946×10338.946\times 10^{33}
Sign 11
Analytic cond. 6.01132×1086.01132\times 10^{8}
Root an. cond. 2.321612.32161
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  − 36·11-s + 24·19-s + 42·29-s − 6·31-s − 36·41-s + 12·49-s + 72·61-s + 2·64-s + 6·71-s + 24·79-s − 6·89-s − 42·101-s + 60·109-s + 702·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 10.8·11-s + 5.50·19-s + 7.79·29-s − 1.07·31-s − 5.62·41-s + 12/7·49-s + 9.21·61-s + 1/4·64-s + 0.712·71-s + 2.70·79-s − 0.635·89-s − 4.17·101-s + 5.74·109-s + 63.8·121-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 − 1.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

Λ(s)=((336⋅524)s/2 ΓC(s)12 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((336⋅524)s/2 ΓC(s+1/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 336⋅5243^{36} \cdot 5^{24}
Sign: 11
Analytic conductor: 6.01132×1086.01132\times 10^{8}
Root analytic conductor: 2.321612.32161
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 336⋅524, ( :[1/2]12), 1)(24,\ 3^{36} \cdot 5^{24} ,\ ( \ : [1/2]^{12} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 5.2162305315.216230531
L(12)L(\frac12) ≈\approx 5.2162305315.216230531
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+p3T6+p6T12 1 + p^{3} T^{6} + p^{6} T^{12}
5 1 1
good2 1−pT6−27T10+21T12−27p2T14−p7T18+p12T24 1 - p T^{6} - 27 T^{10} + 21 T^{12} - 27 p^{2} T^{14} - p^{7} T^{18} + p^{12} T^{24}
7 1−12T2+24T4+262T6−1944T8+8424T10−38205T12+8424p2T14−1944p4T16+262p6T18+24p8T20−12p10T22+p12T24 1 - 12 T^{2} + 24 T^{4} + 262 T^{6} - 1944 T^{8} + 8424 T^{10} - 38205 T^{12} + 8424 p^{2} T^{14} - 1944 p^{4} T^{16} + 262 p^{6} T^{18} + 24 p^{8} T^{20} - 12 p^{10} T^{22} + p^{12} T^{24}
11 (1+18T+135T2+513T3+675T4−3033T5−18422T6−3033pT7+675p2T8+513p3T9+135p4T10+18p5T11+p6T12)2 ( 1 + 18 T + 135 T^{2} + 513 T^{3} + 675 T^{4} - 3033 T^{5} - 18422 T^{6} - 3033 p T^{7} + 675 p^{2} T^{8} + 513 p^{3} T^{9} + 135 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2}
13 1+18T2−261T4−8633T6−40239T8+953919T10+19127910T12+953919p2T14−40239p4T16−8633p6T18−261p8T20+18p10T22+p12T24 1 + 18 T^{2} - 261 T^{4} - 8633 T^{6} - 40239 T^{8} + 953919 T^{10} + 19127910 T^{12} + 953919 p^{2} T^{14} - 40239 p^{4} T^{16} - 8633 p^{6} T^{18} - 261 p^{8} T^{20} + 18 p^{10} T^{22} + p^{12} T^{24}
17 1+84T2+3900T4+127966T6+3288456T8+70189416T10+1282121619T12+70189416p2T14+3288456p4T16+127966p6T18+3900p8T20+84p10T22+p12T24 1 + 84 T^{2} + 3900 T^{4} + 127966 T^{6} + 3288456 T^{8} + 70189416 T^{10} + 1282121619 T^{12} + 70189416 p^{2} T^{14} + 3288456 p^{4} T^{16} + 127966 p^{6} T^{18} + 3900 p^{8} T^{20} + 84 p^{10} T^{22} + p^{12} T^{24}
19 (1−12T+51T2−188T3+1314T4−5076T5+12699T6−5076pT7+1314p2T8−188p3T9+51p4T10−12p5T11+p6T12)2 ( 1 - 12 T + 51 T^{2} - 188 T^{3} + 1314 T^{4} - 5076 T^{5} + 12699 T^{6} - 5076 p T^{7} + 1314 p^{2} T^{8} - 188 p^{3} T^{9} + 51 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2}
23 1−72T2+2628T4−73730T6+1554876T8−25875612T10+500700963T12−25875612p2T14+1554876p4T16−73730p6T18+2628p8T20−72p10T22+p12T24 1 - 72 T^{2} + 2628 T^{4} - 73730 T^{6} + 1554876 T^{8} - 25875612 T^{10} + 500700963 T^{12} - 25875612 p^{2} T^{14} + 1554876 p^{4} T^{16} - 73730 p^{6} T^{18} + 2628 p^{8} T^{20} - 72 p^{10} T^{22} + p^{12} T^{24}
29 (1−21T+207T2−1125T3+1890T4+22776T5−210023T6+22776pT7+1890p2T8−1125p3T9+207p4T10−21p5T11+p6T12)2 ( 1 - 21 T + 207 T^{2} - 1125 T^{3} + 1890 T^{4} + 22776 T^{5} - 210023 T^{6} + 22776 p T^{7} + 1890 p^{2} T^{8} - 1125 p^{3} T^{9} + 207 p^{4} T^{10} - 21 p^{5} T^{11} + p^{6} T^{12} )^{2}
31 (1+3T+24T2+80T3+423T4−4653T5−5787T6−4653pT7+423p2T8+80p3T9+24p4T10+3p5T11+p6T12)2 ( 1 + 3 T + 24 T^{2} + 80 T^{3} + 423 T^{4} - 4653 T^{5} - 5787 T^{6} - 4653 p T^{7} + 423 p^{2} T^{8} + 80 p^{3} T^{9} + 24 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2}
37 1+132T2+8412T4+391870T6+16229592T8+584807544T10+20378076867T12+584807544p2T14+16229592p4T16+391870p6T18+8412p8T20+132p10T22+p12T24 1 + 132 T^{2} + 8412 T^{4} + 391870 T^{6} + 16229592 T^{8} + 584807544 T^{10} + 20378076867 T^{12} + 584807544 p^{2} T^{14} + 16229592 p^{4} T^{16} + 391870 p^{6} T^{18} + 8412 p^{8} T^{20} + 132 p^{10} T^{22} + p^{12} T^{24}
41 (1+18T+171T2+1143T3+9009T4+79119T5+597574T6+79119pT7+9009p2T8+1143p3T9+171p4T10+18p5T11+p6T12)2 ( 1 + 18 T + 171 T^{2} + 1143 T^{3} + 9009 T^{4} + 79119 T^{5} + 597574 T^{6} + 79119 p T^{7} + 9009 p^{2} T^{8} + 1143 p^{3} T^{9} + 171 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2}
43 1−276T2+40134T4−3953309T6+292635369T8−17126403723T10+814017920361T12−17126403723p2T14+292635369p4T16−3953309p6T18+40134p8T20−276p10T22+p12T24 1 - 276 T^{2} + 40134 T^{4} - 3953309 T^{6} + 292635369 T^{8} - 17126403723 T^{10} + 814017920361 T^{12} - 17126403723 p^{2} T^{14} + 292635369 p^{4} T^{16} - 3953309 p^{6} T^{18} + 40134 p^{8} T^{20} - 276 p^{10} T^{22} + p^{12} T^{24}
47 1+36T2+2574T4−23141T6−166887T8+70522893T10+7143449961T12+70522893p2T14−166887p4T16−23141p6T18+2574p8T20+36p10T22+p12T24 1 + 36 T^{2} + 2574 T^{4} - 23141 T^{6} - 166887 T^{8} + 70522893 T^{10} + 7143449961 T^{12} + 70522893 p^{2} T^{14} - 166887 p^{4} T^{16} - 23141 p^{6} T^{18} + 2574 p^{8} T^{20} + 36 p^{10} T^{22} + p^{12} T^{24}
53 (1−84T2+6072T4−362545T6+6072p2T8−84p4T10+p6T12)2 ( 1 - 84 T^{2} + 6072 T^{4} - 362545 T^{6} + 6072 p^{2} T^{8} - 84 p^{4} T^{10} + p^{6} T^{12} )^{2}
59 (1+144T2−576T3+144pT4−97596T5+406765T6−97596pT7+144p3T8−576p3T9+144p4T10+p6T12)2 ( 1 + 144 T^{2} - 576 T^{3} + 144 p T^{4} - 97596 T^{5} + 406765 T^{6} - 97596 p T^{7} + 144 p^{3} T^{8} - 576 p^{3} T^{9} + 144 p^{4} T^{10} + p^{6} T^{12} )^{2}
61 (1−36T+756T2−190pT3+141336T4−1424232T5+12082719T6−1424232pT7+141336p2T8−190p4T9+756p4T10−36p5T11+p6T12)2 ( 1 - 36 T + 756 T^{2} - 190 p T^{3} + 141336 T^{4} - 1424232 T^{5} + 12082719 T^{6} - 1424232 p T^{7} + 141336 p^{2} T^{8} - 190 p^{4} T^{9} + 756 p^{4} T^{10} - 36 p^{5} T^{11} + p^{6} T^{12} )^{2}
67 1−144T2+11844T4−912746T6+87406884T8−6957232452T10+499884420891T12−6957232452p2T14+87406884p4T16−912746p6T18+11844p8T20−144p10T22+p12T24 1 - 144 T^{2} + 11844 T^{4} - 912746 T^{6} + 87406884 T^{8} - 6957232452 T^{10} + 499884420891 T^{12} - 6957232452 p^{2} T^{14} + 87406884 p^{4} T^{16} - 912746 p^{6} T^{18} + 11844 p^{8} T^{20} - 144 p^{10} T^{22} + p^{12} T^{24}
71 (1−3T−150T2−63T3+13371T4+19698T5−1081649T6+19698pT7+13371p2T8−63p3T9−150p4T10−3p5T11+p6T12)2 ( 1 - 3 T - 150 T^{2} - 63 T^{3} + 13371 T^{4} + 19698 T^{5} - 1081649 T^{6} + 19698 p T^{7} + 13371 p^{2} T^{8} - 63 p^{3} T^{9} - 150 p^{4} T^{10} - 3 p^{5} T^{11} + p^{6} T^{12} )^{2}
73 1+408T2+95268T4+15399598T6+1905199452T8+187847914644T10+15132874130115T12+187847914644p2T14+1905199452p4T16+15399598p6T18+95268p8T20+408p10T22+p12T24 1 + 408 T^{2} + 95268 T^{4} + 15399598 T^{6} + 1905199452 T^{8} + 187847914644 T^{10} + 15132874130115 T^{12} + 187847914644 p^{2} T^{14} + 1905199452 p^{4} T^{16} + 15399598 p^{6} T^{18} + 95268 p^{8} T^{20} + 408 p^{10} T^{22} + p^{12} T^{24}
79 (1−12T+204T2−2915T3+31545T4−326493T5+3330513T6−326493pT7+31545p2T8−2915p3T9+204p4T10−12p5T11+p6T12)2 ( 1 - 12 T + 204 T^{2} - 2915 T^{3} + 31545 T^{4} - 326493 T^{5} + 3330513 T^{6} - 326493 p T^{7} + 31545 p^{2} T^{8} - 2915 p^{3} T^{9} + 204 p^{4} T^{10} - 12 p^{5} T^{11} + p^{6} T^{12} )^{2}
83 1+468T2+93294T4+9694123T6+419139873T8−21991579731T10−4161640619895T12−21991579731p2T14+419139873p4T16+9694123p6T18+93294p8T20+468p10T22+p12T24 1 + 468 T^{2} + 93294 T^{4} + 9694123 T^{6} + 419139873 T^{8} - 21991579731 T^{10} - 4161640619895 T^{12} - 21991579731 p^{2} T^{14} + 419139873 p^{4} T^{16} + 9694123 p^{6} T^{18} + 93294 p^{8} T^{20} + 468 p^{10} T^{22} + p^{12} T^{24}
89 (1+3T−186T2−585T3+18915T4+33708T5−1703351T6+33708pT7+18915p2T8−585p3T9−186p4T10+3p5T11+p6T12)2 ( 1 + 3 T - 186 T^{2} - 585 T^{3} + 18915 T^{4} + 33708 T^{5} - 1703351 T^{6} + 33708 p T^{7} + 18915 p^{2} T^{8} - 585 p^{3} T^{9} - 186 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2}
97 1+93T2−1770T4−1760936T6−103886055T8+10655050167T10+2102618450697T12+10655050167p2T14−103886055p4T16−1760936p6T18−1770p8T20+93p10T22+p12T24 1 + 93 T^{2} - 1770 T^{4} - 1760936 T^{6} - 103886055 T^{8} + 10655050167 T^{10} + 2102618450697 T^{12} + 10655050167 p^{2} T^{14} - 103886055 p^{4} T^{16} - 1760936 p^{6} T^{18} - 1770 p^{8} T^{20} + 93 p^{10} T^{22} + p^{12} T^{24}
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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

−3.33252427444800281087361549331, −3.32583475421166310092946574270, −3.25350824375078114235048697771, −3.05328623206781612081605159449, −2.97443540934827751570052927838, −2.85449468057580580454719898934, −2.70252703236316582122298059914, −2.66085796499277951586985337821, −2.59296822880133799517056746840, −2.50679298021494174889355704822, −2.45189780804244914999831040983, −2.40108414581629192260220702580, −2.32169170507917870786312608845, −2.24135412528041975636787624262, −2.08399801724437977880282913809, −1.79262514607155529390763129522, −1.71934796596379884188946755897, −1.48445222708352362642519633307, −1.21192609207470900952189099355, −1.14658906272257958365919149968, −0.77684381566177173291517508519, −0.62849322601942629842673597837, −0.61989409660249548682358214669, −0.49223401117379135420759553065, −0.38994917036326733227311862101, 0.38994917036326733227311862101, 0.49223401117379135420759553065, 0.61989409660249548682358214669, 0.62849322601942629842673597837, 0.77684381566177173291517508519, 1.14658906272257958365919149968, 1.21192609207470900952189099355, 1.48445222708352362642519633307, 1.71934796596379884188946755897, 1.79262514607155529390763129522, 2.08399801724437977880282913809, 2.24135412528041975636787624262, 2.32169170507917870786312608845, 2.40108414581629192260220702580, 2.45189780804244914999831040983, 2.50679298021494174889355704822, 2.59296822880133799517056746840, 2.66085796499277951586985337821, 2.70252703236316582122298059914, 2.85449468057580580454719898934, 2.97443540934827751570052927838, 3.05328623206781612081605159449, 3.25350824375078114235048697771, 3.32583475421166310092946574270, 3.33252427444800281087361549331

Graph of the ZZ-function along the critical line

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