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Properties

Label 32-44e16-1.1-c1e16-0-0
Degree 3232
Conductor 1.974×10261.974\times 10^{26}
Sign 11
Analytic cond. 5.39108×10−85.39108\times 10^{-8}
Root an. cond. 0.5927400.592740
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  − 5·2-s + 12·4-s − 6·5-s − 20·8-s − 11·9-s + 30·10-s − 10·13-s + 34·16-s − 10·17-s + 55·18-s − 72·20-s + 31·25-s + 50·26-s − 10·29-s − 65·32-s + 50·34-s − 132·36-s + 18·37-s + 120·40-s + 10·41-s + 66·45-s + 17·49-s − 155·50-s − 120·52-s + 38·53-s + 50·58-s − 10·61-s + ⋯
L(s)  = 1  − 3.53·2-s + 6·4-s − 2.68·5-s − 7.07·8-s − 3.66·9-s + 9.48·10-s − 2.77·13-s + 17/2·16-s − 2.42·17-s + 12.9·18-s − 16.0·20-s + 31/5·25-s + 9.80·26-s − 1.85·29-s − 11.4·32-s + 8.57·34-s − 22·36-s + 2.95·37-s + 18.9·40-s + 1.56·41-s + 9.83·45-s + 17/7·49-s − 21.9·50-s − 16.6·52-s + 5.21·53-s + 6.56·58-s − 1.28·61-s + ⋯

Functional equation

Λ(s)=((232⋅1116)s/2 ΓC(s)16 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((232⋅1116)s/2 ΓC(s+1/2)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 232⋅11162^{32} \cdot 11^{16}
Sign: 11
Analytic conductor: 5.39108×10−85.39108\times 10^{-8}
Root analytic conductor: 0.5927400.592740
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 232⋅1116, ( :[1/2]16), 1)(32,\ 2^{32} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.0032192401110.003219240111
L(12)L(\frac12) ≈\approx 0.0032192401110.003219240111
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+5T+13T2+25T3+35T4+15pT5−pT6−15p2T7−29p2T8−15p3T9−p3T10+15p4T11+35p4T12+25p5T13+13p6T14+5p7T15+p8T16 1 + 5 T + 13 T^{2} + 25 T^{3} + 35 T^{4} + 15 p T^{5} - p T^{6} - 15 p^{2} T^{7} - 29 p^{2} T^{8} - 15 p^{3} T^{9} - p^{3} T^{10} + 15 p^{4} T^{11} + 35 p^{4} T^{12} + 25 p^{5} T^{13} + 13 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16}
11 1−35T2+519T4−5365T6+5136pT8−5365p2T10+519p4T12−35p6T14+p8T16 1 - 35 T^{2} + 519 T^{4} - 5365 T^{6} + 5136 p T^{8} - 5365 p^{2} T^{10} + 519 p^{4} T^{12} - 35 p^{6} T^{14} + p^{8} T^{16}
good3 1+11T2+2p3T4+176T6+2p5T8+491pT10+659p2T12+23828T14+78832T16+23828p2T18+659p6T20+491p7T22+2p13T24+176p10T26+2p15T28+11p14T30+p16T32 1 + 11 T^{2} + 2 p^{3} T^{4} + 176 T^{6} + 2 p^{5} T^{8} + 491 p T^{10} + 659 p^{2} T^{12} + 23828 T^{14} + 78832 T^{16} + 23828 p^{2} T^{18} + 659 p^{6} T^{20} + 491 p^{7} T^{22} + 2 p^{13} T^{24} + 176 p^{10} T^{26} + 2 p^{15} T^{28} + 11 p^{14} T^{30} + p^{16} T^{32}
5 (1+3T−2T2−14T3−2pT4+51T5+153T6−252T7−1564T8−252pT9+153p2T10+51p3T11−2p5T12−14p5T13−2p6T14+3p7T15+p8T16)2 ( 1 + 3 T - 2 T^{2} - 14 T^{3} - 2 p T^{4} + 51 T^{5} + 153 T^{6} - 252 T^{7} - 1564 T^{8} - 252 p T^{9} + 153 p^{2} T^{10} + 51 p^{3} T^{11} - 2 p^{5} T^{12} - 14 p^{5} T^{13} - 2 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} )^{2}
7 1−17T2+170T4−160pT6+3590T8−5391T10+51607T12−1265760T14+13240480T16−1265760p2T18+51607p4T20−5391p6T22+3590p8T24−160p11T26+170p12T28−17p14T30+p16T32 1 - 17 T^{2} + 170 T^{4} - 160 p T^{6} + 3590 T^{8} - 5391 T^{10} + 51607 T^{12} - 1265760 T^{14} + 13240480 T^{16} - 1265760 p^{2} T^{18} + 51607 p^{4} T^{20} - 5391 p^{6} T^{22} + 3590 p^{8} T^{24} - 160 p^{11} T^{26} + 170 p^{12} T^{28} - 17 p^{14} T^{30} + p^{16} T^{32}
13 (1+5T+42T2+150T3+620T4+1455T5+387T6−3650T7−65036T8−3650pT9+387p2T10+1455p3T11+620p4T12+150p5T13+42p6T14+5p7T15+p8T16)2 ( 1 + 5 T + 42 T^{2} + 150 T^{3} + 620 T^{4} + 1455 T^{5} + 387 T^{6} - 3650 T^{7} - 65036 T^{8} - 3650 p T^{9} + 387 p^{2} T^{10} + 1455 p^{3} T^{11} + 620 p^{4} T^{12} + 150 p^{5} T^{13} + 42 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2}
17 (1+5T+10T2+486T4+2555T5+10495T6+25600T7+143056T8+25600pT9+10495p2T10+2555p3T11+486p4T12+10p6T14+5p7T15+p8T16)2 ( 1 + 5 T + 10 T^{2} + 486 T^{4} + 2555 T^{5} + 10495 T^{6} + 25600 T^{7} + 143056 T^{8} + 25600 p T^{9} + 10495 p^{2} T^{10} + 2555 p^{3} T^{11} + 486 p^{4} T^{12} + 10 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2}
19 1+30T2+1233T4+23510T6+411393T8+1849230T10−90963349T12−3587444450T14−90905039420T16−3587444450p2T18−90963349p4T20+1849230p6T22+411393p8T24+23510p10T26+1233p12T28+30p14T30+p16T32 1 + 30 T^{2} + 1233 T^{4} + 23510 T^{6} + 411393 T^{8} + 1849230 T^{10} - 90963349 T^{12} - 3587444450 T^{14} - 90905039420 T^{16} - 3587444450 p^{2} T^{18} - 90963349 p^{4} T^{20} + 1849230 p^{6} T^{22} + 411393 p^{8} T^{24} + 23510 p^{10} T^{26} + 1233 p^{12} T^{28} + 30 p^{14} T^{30} + p^{16} T^{32}
23 (1−96T2+4732T4−162208T6+4234310T8−162208p2T10+4732p4T12−96p6T14+p8T16)2 ( 1 - 96 T^{2} + 4732 T^{4} - 162208 T^{6} + 4234310 T^{8} - 162208 p^{2} T^{10} + 4732 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} )^{2}
29 (1+5T+78T2+550T3+3768T4+26695T5+166131T6+945350T7+5661780T8+945350pT9+166131p2T10+26695p3T11+3768p4T12+550p5T13+78p6T14+5p7T15+p8T16)2 ( 1 + 5 T + 78 T^{2} + 550 T^{3} + 3768 T^{4} + 26695 T^{5} + 166131 T^{6} + 945350 T^{7} + 5661780 T^{8} + 945350 p T^{9} + 166131 p^{2} T^{10} + 26695 p^{3} T^{11} + 3768 p^{4} T^{12} + 550 p^{5} T^{13} + 78 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2}
31 1+13T2+46T4−39168T6−244734T8+28347139T10+776025319T12−11750120144T14−1558123543328T16−11750120144p2T18+776025319p4T20+28347139p6T22−244734p8T24−39168p10T26+46p12T28+13p14T30+p16T32 1 + 13 T^{2} + 46 T^{4} - 39168 T^{6} - 244734 T^{8} + 28347139 T^{10} + 776025319 T^{12} - 11750120144 T^{14} - 1558123543328 T^{16} - 11750120144 p^{2} T^{18} + 776025319 p^{4} T^{20} + 28347139 p^{6} T^{22} - 244734 p^{8} T^{24} - 39168 p^{10} T^{26} + 46 p^{12} T^{28} + 13 p^{14} T^{30} + p^{16} T^{32}
37 (1−9T+58T2−288T3+2826T4−8739T5+5135T6+158436T7+832784T8+158436pT9+5135p2T10−8739p3T11+2826p4T12−288p5T13+58p6T14−9p7T15+p8T16)2 ( 1 - 9 T + 58 T^{2} - 288 T^{3} + 2826 T^{4} - 8739 T^{5} + 5135 T^{6} + 158436 T^{7} + 832784 T^{8} + 158436 p T^{9} + 5135 p^{2} T^{10} - 8739 p^{3} T^{11} + 2826 p^{4} T^{12} - 288 p^{5} T^{13} + 58 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} )^{2}
41 (1−5T+8T2−540T3+2928T4−5515T5+91991T6−523800T7+1936120T8−523800pT9+91991p2T10−5515p3T11+2928p4T12−540p5T13+8p6T14−5p7T15+p8T16)2 ( 1 - 5 T + 8 T^{2} - 540 T^{3} + 2928 T^{4} - 5515 T^{5} + 91991 T^{6} - 523800 T^{7} + 1936120 T^{8} - 523800 p T^{9} + 91991 p^{2} T^{10} - 5515 p^{3} T^{11} + 2928 p^{4} T^{12} - 540 p^{5} T^{13} + 8 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} )^{2}
43 (1+201T2+20307T4+1361163T6+67236960T8+1361163p2T10+20307p4T12+201p6T14+p8T16)2 ( 1 + 201 T^{2} + 20307 T^{4} + 1361163 T^{6} + 67236960 T^{8} + 1361163 p^{2} T^{10} + 20307 p^{4} T^{12} + 201 p^{6} T^{14} + p^{8} T^{16} )^{2}
47 1+59T2+1174T4−129116T6−4850414T8+212153137T10+11841271291T12−458606392528T14−66560830358008T16−458606392528p2T18+11841271291p4T20+212153137p6T22−4850414p8T24−129116p10T26+1174p12T28+59p14T30+p16T32 1 + 59 T^{2} + 1174 T^{4} - 129116 T^{6} - 4850414 T^{8} + 212153137 T^{10} + 11841271291 T^{12} - 458606392528 T^{14} - 66560830358008 T^{16} - 458606392528 p^{2} T^{18} + 11841271291 p^{4} T^{20} + 212153137 p^{6} T^{22} - 4850414 p^{8} T^{24} - 129116 p^{10} T^{26} + 1174 p^{12} T^{28} + 59 p^{14} T^{30} + p^{16} T^{32}
53 (1−19T+56T2+1024T3−7582T4+24781T5−179211T6−1855738T7+36671448T8−1855738pT9−179211p2T10+24781p3T11−7582p4T12+1024p5T13+56p6T14−19p7T15+p8T16)2 ( 1 - 19 T + 56 T^{2} + 1024 T^{3} - 7582 T^{4} + 24781 T^{5} - 179211 T^{6} - 1855738 T^{7} + 36671448 T^{8} - 1855738 p T^{9} - 179211 p^{2} T^{10} + 24781 p^{3} T^{11} - 7582 p^{4} T^{12} + 1024 p^{5} T^{13} + 56 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} )^{2}
59 1+350T2+56233T4+5299950T6+306491273T8+9575397350T10−4883294509T12−17205708063250T14−1208178268282700T16−17205708063250p2T18−4883294509p4T20+9575397350p6T22+306491273p8T24+5299950p10T26+56233p12T28+350p14T30+p16T32 1 + 350 T^{2} + 56233 T^{4} + 5299950 T^{6} + 306491273 T^{8} + 9575397350 T^{10} - 4883294509 T^{12} - 17205708063250 T^{14} - 1208178268282700 T^{16} - 17205708063250 p^{2} T^{18} - 4883294509 p^{4} T^{20} + 9575397350 p^{6} T^{22} + 306491273 p^{8} T^{24} + 5299950 p^{10} T^{26} + 56233 p^{12} T^{28} + 350 p^{14} T^{30} + p^{16} T^{32}
61 (1+5T−4T2+340T3+6550T4+21685T5+106569T6+2625650T7+27744944T8+2625650pT9+106569p2T10+21685p3T11+6550p4T12+340p5T13−4p6T14+5p7T15+p8T16)2 ( 1 + 5 T - 4 T^{2} + 340 T^{3} + 6550 T^{4} + 21685 T^{5} + 106569 T^{6} + 2625650 T^{7} + 27744944 T^{8} + 2625650 p T^{9} + 106569 p^{2} T^{10} + 21685 p^{3} T^{11} + 6550 p^{4} T^{12} + 340 p^{5} T^{13} - 4 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} )^{2}
67 (1−351T2+61547T4−6900053T6+544631520T8−6900053p2T10+61547p4T12−351p6T14+p8T16)2 ( 1 - 351 T^{2} + 61547 T^{4} - 6900053 T^{6} + 544631520 T^{8} - 6900053 p^{2} T^{10} + 61547 p^{4} T^{12} - 351 p^{6} T^{14} + p^{8} T^{16} )^{2}
71 1+285T2+39458T4+3119420T6+118131358T8+404779135T10+205412628291T12+73614429095400T14+7731123817186280T16+73614429095400p2T18+205412628291p4T20+404779135p6T22+118131358p8T24+3119420p10T26+39458p12T28+285p14T30+p16T32 1 + 285 T^{2} + 39458 T^{4} + 3119420 T^{6} + 118131358 T^{8} + 404779135 T^{10} + 205412628291 T^{12} + 73614429095400 T^{14} + 7731123817186280 T^{16} + 73614429095400 p^{2} T^{18} + 205412628291 p^{4} T^{20} + 404779135 p^{6} T^{22} + 118131358 p^{8} T^{24} + 3119420 p^{10} T^{26} + 39458 p^{12} T^{28} + 285 p^{14} T^{30} + p^{16} T^{32}
73 (1+15T+182T2+960T3+8490T4+51585T5+875047T6+4632960T7+56899184T8+4632960pT9+875047p2T10+51585p3T11+8490p4T12+960p5T13+182p6T14+15p7T15+p8T16)2 ( 1 + 15 T + 182 T^{2} + 960 T^{3} + 8490 T^{4} + 51585 T^{5} + 875047 T^{6} + 4632960 T^{7} + 56899184 T^{8} + 4632960 p T^{9} + 875047 p^{2} T^{10} + 51585 p^{3} T^{11} + 8490 p^{4} T^{12} + 960 p^{5} T^{13} + 182 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} )^{2}
79 1−175T2+25018T4−2721680T6+276506938T8−24540773525T10+2169836003371T12−177413374417900T14+14750049423098000T16−177413374417900p2T18+2169836003371p4T20−24540773525p6T22+276506938p8T24−2721680p10T26+25018p12T28−175p14T30+p16T32 1 - 175 T^{2} + 25018 T^{4} - 2721680 T^{6} + 276506938 T^{8} - 24540773525 T^{10} + 2169836003371 T^{12} - 177413374417900 T^{14} + 14750049423098000 T^{16} - 177413374417900 p^{2} T^{18} + 2169836003371 p^{4} T^{20} - 24540773525 p^{6} T^{22} + 276506938 p^{8} T^{24} - 2721680 p^{10} T^{26} + 25018 p^{12} T^{28} - 175 p^{14} T^{30} + p^{16} T^{32}
83 1−432T2+87375T4−11521130T6+1183958625T8−112754321886T10+11335103772317T12−1139715862795000T14+102156150284526820T16−1139715862795000p2T18+11335103772317p4T20−112754321886p6T22+1183958625p8T24−11521130p10T26+87375p12T28−432p14T30+p16T32 1 - 432 T^{2} + 87375 T^{4} - 11521130 T^{6} + 1183958625 T^{8} - 112754321886 T^{10} + 11335103772317 T^{12} - 1139715862795000 T^{14} + 102156150284526820 T^{16} - 1139715862795000 p^{2} T^{18} + 11335103772317 p^{4} T^{20} - 112754321886 p^{6} T^{22} + 1183958625 p^{8} T^{24} - 11521130 p^{10} T^{26} + 87375 p^{12} T^{28} - 432 p^{14} T^{30} + p^{16} T^{32}
89 (1+9T+337T2+2217T3+44260T4+2217pT5+337p2T6+9p3T7+p4T8)4 ( 1 + 9 T + 337 T^{2} + 2217 T^{3} + 44260 T^{4} + 2217 p T^{5} + 337 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} )^{4}
97 (1+17T+87T2+1195T3+20696T4+1195pT5+87p2T6+17p3T7+p4T8)4 ( 1 + 17 T + 87 T^{2} + 1195 T^{3} + 20696 T^{4} + 1195 p T^{5} + 87 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} )^{4}
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   L(s)=∏p ∏j=132(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−5.48796192545704803665547124242, −5.42921536950679246297502834054, −5.38726880851750830540119245571, −5.02909577387759671992870111748, −4.98840083919941029063903722522, −4.77211385045028668663464353930, −4.65681933478518202795766550077, −4.60794759224348415695938603222, −4.54091099639989253225350124466, −4.21844425640318850237605637302, −4.13958081293968062323330834959, −4.12555725224630642994296772726, −3.88494080891157511628563271071, −3.77428715166864530891938463609, −3.76763754896625893689228736800, −3.29789081753004457615719220340, −3.20137025099284401050165051776, −3.16150572257945797452774858010, −2.76082777466018957972996401558, −2.61795708095079875935400015112, −2.60254900193392084902177184120, −2.58412155340676958706680891206, −2.43970731718250358503347518610, −2.26086986162037297193259723631, −1.30057736695860585162283973664, 1.30057736695860585162283973664, 2.26086986162037297193259723631, 2.43970731718250358503347518610, 2.58412155340676958706680891206, 2.60254900193392084902177184120, 2.61795708095079875935400015112, 2.76082777466018957972996401558, 3.16150572257945797452774858010, 3.20137025099284401050165051776, 3.29789081753004457615719220340, 3.76763754896625893689228736800, 3.77428715166864530891938463609, 3.88494080891157511628563271071, 4.12555725224630642994296772726, 4.13958081293968062323330834959, 4.21844425640318850237605637302, 4.54091099639989253225350124466, 4.60794759224348415695938603222, 4.65681933478518202795766550077, 4.77211385045028668663464353930, 4.98840083919941029063903722522, 5.02909577387759671992870111748, 5.38726880851750830540119245571, 5.42921536950679246297502834054, 5.48796192545704803665547124242

Graph of the ZZ-function along the critical line

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