↑

Properties

Label 32-55e16-1.1-c1e16-0-0
Degree 3232
Conductor 7.011×10277.011\times 10^{27}
Sign 11
Analytic cond. 1.91529×10−61.91529\times 10^{-6}
Root an. cond. 0.6627040.662704
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·4-s − 2·5-s − 5·9-s − 6·11-s + 25·16-s + 6·19-s + 12·20-s − 6·25-s + 2·29-s + 8·31-s + 30·36-s − 52·41-s + 36·44-s + 10·45-s − 19·49-s + 12·55-s + 2·59-s − 40·61-s − 74·64-s + 36·71-s − 36·76-s + 38·79-s − 50·80-s + 31·81-s + 24·89-s − 12·95-s + 30·99-s + ⋯
L(s)  = 1  − 3·4-s − 0.894·5-s − 5/3·9-s − 1.80·11-s + 25/4·16-s + 1.37·19-s + 2.68·20-s − 6/5·25-s + 0.371·29-s + 1.43·31-s + 5·36-s − 8.12·41-s + 5.42·44-s + 1.49·45-s − 2.71·49-s + 1.61·55-s + 0.260·59-s − 5.12·61-s − 9.25·64-s + 4.27·71-s − 4.12·76-s + 4.27·79-s − 5.59·80-s + 31/9·81-s + 2.54·89-s − 1.23·95-s + 3.01·99-s + ⋯

Functional equation

Λ(s)=((516⋅1116)s/2 ΓC(s)16 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((516⋅1116)s/2 ΓC(s+1/2)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16} \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: 516⋅11165^{16} \cdot 11^{16}
Sign: 11
Analytic conductor: 1.91529×10−61.91529\times 10^{-6}
Root analytic conductor: 0.6627040.662704
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 516⋅1116, ( :[1/2]16), 1)(32,\ 5^{16} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 0.037263167190.03726316719
L(12)L(\frac12) ≈\approx 0.037263167190.03726316719
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)
bad5 1+2T+2pT2+42T3+99T4+338T5+36p2T6+2008T7+5281T8+2008pT9+36p4T10+338p3T11+99p4T12+42p5T13+2p7T14+2p7T15+p8T16 1 + 2 T + 2 p T^{2} + 42 T^{3} + 99 T^{4} + 338 T^{5} + 36 p^{2} T^{6} + 2008 T^{7} + 5281 T^{8} + 2008 p T^{9} + 36 p^{4} T^{10} + 338 p^{3} T^{11} + 99 p^{4} T^{12} + 42 p^{5} T^{13} + 2 p^{7} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16}
11 (1+3T−2pT2−19T3+335T4−19pT5−2p3T6+3p3T7+p4T8)2 ( 1 + 3 T - 2 p T^{2} - 19 T^{3} + 335 T^{4} - 19 p T^{5} - 2 p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2}
good2 1+3pT2+11T4−5pT6−77T8−55pT10+69T12+129p2T14+1193T16+129p4T18+69p4T20−55p7T22−77p8T24−5p11T26+11p12T28+3p15T30+p16T32 1 + 3 p T^{2} + 11 T^{4} - 5 p T^{6} - 77 T^{8} - 55 p T^{10} + 69 T^{12} + 129 p^{2} T^{14} + 1193 T^{16} + 129 p^{4} T^{18} + 69 p^{4} T^{20} - 55 p^{7} T^{22} - 77 p^{8} T^{24} - 5 p^{11} T^{26} + 11 p^{12} T^{28} + 3 p^{15} T^{30} + p^{16} T^{32}
3 1+5T2−2pT4−140T6−10p3T8+1165T10+5506T12−3770T14−53741T16−3770p2T18+5506p4T20+1165p6T22−10p11T24−140p10T26−2p13T28+5p14T30+p16T32 1 + 5 T^{2} - 2 p T^{4} - 140 T^{6} - 10 p^{3} T^{8} + 1165 T^{10} + 5506 T^{12} - 3770 T^{14} - 53741 T^{16} - 3770 p^{2} T^{18} + 5506 p^{4} T^{20} + 1165 p^{6} T^{22} - 10 p^{11} T^{24} - 140 p^{10} T^{26} - 2 p^{13} T^{28} + 5 p^{14} T^{30} + p^{16} T^{32}
7 1+19T2+229T4+2109T6+2213pT8+92002T10+438866T12+1927012T14+12254167T16+1927012p2T18+438866p4T20+92002p6T22+2213p9T24+2109p10T26+229p12T28+19p14T30+p16T32 1 + 19 T^{2} + 229 T^{4} + 2109 T^{6} + 2213 p T^{8} + 92002 T^{10} + 438866 T^{12} + 1927012 T^{14} + 12254167 T^{16} + 1927012 p^{2} T^{18} + 438866 p^{4} T^{20} + 92002 p^{6} T^{22} + 2213 p^{9} T^{24} + 2109 p^{10} T^{26} + 229 p^{12} T^{28} + 19 p^{14} T^{30} + p^{16} T^{32}
13 1+42T2+635T4−275T6−168035T8−2770249T10−11553003T12+300462090T14+6477797525T16+300462090p2T18−11553003p4T20−2770249p6T22−168035p8T24−275p10T26+635p12T28+42p14T30+p16T32 1 + 42 T^{2} + 635 T^{4} - 275 T^{6} - 168035 T^{8} - 2770249 T^{10} - 11553003 T^{12} + 300462090 T^{14} + 6477797525 T^{16} + 300462090 p^{2} T^{18} - 11553003 p^{4} T^{20} - 2770249 p^{6} T^{22} - 168035 p^{8} T^{24} - 275 p^{10} T^{26} + 635 p^{12} T^{28} + 42 p^{14} T^{30} + p^{16} T^{32}
17 1−5T2−186T4−10315T6+52680T8+4287860T10+38701866T12−14826280pT14−31527815401T16−14826280p3T18+38701866p4T20+4287860p6T22+52680p8T24−10315p10T26−186p12T28−5p14T30+p16T32 1 - 5 T^{2} - 186 T^{4} - 10315 T^{6} + 52680 T^{8} + 4287860 T^{10} + 38701866 T^{12} - 14826280 p T^{14} - 31527815401 T^{16} - 14826280 p^{3} T^{18} + 38701866 p^{4} T^{20} + 4287860 p^{6} T^{22} + 52680 p^{8} T^{24} - 10315 p^{10} T^{26} - 186 p^{12} T^{28} - 5 p^{14} T^{30} + p^{16} T^{32}
19 (1−3T−35T2+180T3+185T4−4194T5+14942T6+37935T7−448355T8+37935pT9+14942p2T10−4194p3T11+185p4T12+180p5T13−35p6T14−3p7T15+p8T16)2 ( 1 - 3 T - 35 T^{2} + 180 T^{3} + 185 T^{4} - 4194 T^{5} + 14942 T^{6} + 37935 T^{7} - 448355 T^{8} + 37935 p T^{9} + 14942 p^{2} T^{10} - 4194 p^{3} T^{11} + 185 p^{4} T^{12} + 180 p^{5} T^{13} - 35 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} )^{2}
23 (1−85T2+3382T4−84765T6+1856153T8−84765p2T10+3382p4T12−85p6T14+p8T16)2 ( 1 - 85 T^{2} + 3382 T^{4} - 84765 T^{6} + 1856153 T^{8} - 84765 p^{2} T^{10} + 3382 p^{4} T^{12} - 85 p^{6} T^{14} + p^{8} T^{16} )^{2}
29 (1−T−33T2+278T3+1127T4+610T5−18852T6+40501T7+2275055T8+40501pT9−18852p2T10+610p3T11+1127p4T12+278p5T13−33p6T14−p7T15+p8T16)2 ( 1 - T - 33 T^{2} + 278 T^{3} + 1127 T^{4} + 610 T^{5} - 18852 T^{6} + 40501 T^{7} + 2275055 T^{8} + 40501 p T^{9} - 18852 p^{2} T^{10} + 610 p^{3} T^{11} + 1127 p^{4} T^{12} + 278 p^{5} T^{13} - 33 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} )^{2}
31 (1−4T−23T2−101T3+2551T4−4925T5−57287T6−61630T7+3347567T8−61630pT9−57287p2T10−4925p3T11+2551p4T12−101p5T13−23p6T14−4p7T15+p8T16)2 ( 1 - 4 T - 23 T^{2} - 101 T^{3} + 2551 T^{4} - 4925 T^{5} - 57287 T^{6} - 61630 T^{7} + 3347567 T^{8} - 61630 p T^{9} - 57287 p^{2} T^{10} - 4925 p^{3} T^{11} + 2551 p^{4} T^{12} - 101 p^{5} T^{13} - 23 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2}
37 1+120T2+6419T4+169710T6+1135850T8−4332930pT10−10706145719T12−404428360410T14−12797968713881T16−404428360410p2T18−10706145719p4T20−4332930p7T22+1135850p8T24+169710p10T26+6419p12T28+120p14T30+p16T32 1 + 120 T^{2} + 6419 T^{4} + 169710 T^{6} + 1135850 T^{8} - 4332930 p T^{10} - 10706145719 T^{12} - 404428360410 T^{14} - 12797968713881 T^{16} - 404428360410 p^{2} T^{18} - 10706145719 p^{4} T^{20} - 4332930 p^{7} T^{22} + 1135850 p^{8} T^{24} + 169710 p^{10} T^{26} + 6419 p^{12} T^{28} + 120 p^{14} T^{30} + p^{16} T^{32}
41 (1+26T+263T2+1156T3−1228T4−54770T5−8553pT6−226372T7+7159995T8−226372pT9−8553p3T10−54770p3T11−1228p4T12+1156p5T13+263p6T14+26p7T15+p8T16)2 ( 1 + 26 T + 263 T^{2} + 1156 T^{3} - 1228 T^{4} - 54770 T^{5} - 8553 p T^{6} - 226372 T^{7} + 7159995 T^{8} - 226372 p T^{9} - 8553 p^{3} T^{10} - 54770 p^{3} T^{11} - 1228 p^{4} T^{12} + 1156 p^{5} T^{13} + 263 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} )^{2}
43 (1−171T2+15457T4−1005398T6+49870565T8−1005398p2T10+15457p4T12−171p6T14+p8T16)2 ( 1 - 171 T^{2} + 15457 T^{4} - 1005398 T^{6} + 49870565 T^{8} - 1005398 p^{2} T^{10} + 15457 p^{4} T^{12} - 171 p^{6} T^{14} + p^{8} T^{16} )^{2}
47 1+210T2+23017T4+1971945T6+143287783T8+9109735985T10+538571627359T12+29130862463950T14+1425554876818805T16+29130862463950p2T18+538571627359p4T20+9109735985p6T22+143287783p8T24+1971945p10T26+23017p12T28+210p14T30+p16T32 1 + 210 T^{2} + 23017 T^{4} + 1971945 T^{6} + 143287783 T^{8} + 9109735985 T^{10} + 538571627359 T^{12} + 29130862463950 T^{14} + 1425554876818805 T^{16} + 29130862463950 p^{2} T^{18} + 538571627359 p^{4} T^{20} + 9109735985 p^{6} T^{22} + 143287783 p^{8} T^{24} + 1971945 p^{10} T^{26} + 23017 p^{12} T^{28} + 210 p^{14} T^{30} + p^{16} T^{32}
53 1−5T2−981T4−139345T6−1908885T8+84123170T10+22219415316T12−356460233840T14−33514746895621T16−356460233840p2T18+22219415316p4T20+84123170p6T22−1908885p8T24−139345p10T26−981p12T28−5p14T30+p16T32 1 - 5 T^{2} - 981 T^{4} - 139345 T^{6} - 1908885 T^{8} + 84123170 T^{10} + 22219415316 T^{12} - 356460233840 T^{14} - 33514746895621 T^{16} - 356460233840 p^{2} T^{18} + 22219415316 p^{4} T^{20} + 84123170 p^{6} T^{22} - 1908885 p^{8} T^{24} - 139345 p^{10} T^{26} - 981 p^{12} T^{28} - 5 p^{14} T^{30} + p^{16} T^{32}
59 (1−T+11T2+85T3+7113T4+12500T5−215226T6+1526554T7+21070553T8+1526554pT9−215226p2T10+12500p3T11+7113p4T12+85p5T13+11p6T14−p7T15+p8T16)2 ( 1 - T + 11 T^{2} + 85 T^{3} + 7113 T^{4} + 12500 T^{5} - 215226 T^{6} + 1526554 T^{7} + 21070553 T^{8} + 1526554 p T^{9} - 215226 p^{2} T^{10} + 12500 p^{3} T^{11} + 7113 p^{4} T^{12} + 85 p^{5} T^{13} + 11 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} )^{2}
61 (1+20T+26T2−2140T3−12685T4+139300T5+1704084T6−3518720T7−128652411T8−3518720pT9+1704084p2T10+139300p3T11−12685p4T12−2140p5T13+26p6T14+20p7T15+p8T16)2 ( 1 + 20 T + 26 T^{2} - 2140 T^{3} - 12685 T^{4} + 139300 T^{5} + 1704084 T^{6} - 3518720 T^{7} - 128652411 T^{8} - 3518720 p T^{9} + 1704084 p^{2} T^{10} + 139300 p^{3} T^{11} - 12685 p^{4} T^{12} - 2140 p^{5} T^{13} + 26 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2}
67 (1−387T2+71530T4−8300957T6+662638709T8−8300957p2T10+71530p4T12−387p6T14+p8T16)2 ( 1 - 387 T^{2} + 71530 T^{4} - 8300957 T^{6} + 662638709 T^{8} - 8300957 p^{2} T^{10} + 71530 p^{4} T^{12} - 387 p^{6} T^{14} + p^{8} T^{16} )^{2}
71 (1−18T+119T2−660T3+6528T4−48510T5+528511T6−8889912T7+95792243T8−8889912pT9+528511p2T10−48510p3T11+6528p4T12−660p5T13+119p6T14−18p7T15+p8T16)2 ( 1 - 18 T + 119 T^{2} - 660 T^{3} + 6528 T^{4} - 48510 T^{5} + 528511 T^{6} - 8889912 T^{7} + 95792243 T^{8} - 8889912 p T^{9} + 528511 p^{2} T^{10} - 48510 p^{3} T^{11} + 6528 p^{4} T^{12} - 660 p^{5} T^{13} + 119 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16} )^{2}
73 1+91T2−1406T4−1338589T6−44180104T8+6397599698T10+733001427326T12−18973235763482T14−4254917733021293T16−18973235763482p2T18+733001427326p4T20+6397599698p6T22−44180104p8T24−1338589p10T26−1406p12T28+91p14T30+p16T32 1 + 91 T^{2} - 1406 T^{4} - 1338589 T^{6} - 44180104 T^{8} + 6397599698 T^{10} + 733001427326 T^{12} - 18973235763482 T^{14} - 4254917733021293 T^{16} - 18973235763482 p^{2} T^{18} + 733001427326 p^{4} T^{20} + 6397599698 p^{6} T^{22} - 44180104 p^{8} T^{24} - 1338589 p^{10} T^{26} - 1406 p^{12} T^{28} + 91 p^{14} T^{30} + p^{16} T^{32}
79 (1−19T+52T2+772T3+1222T4−44105T5−607892T6+7482254T7−37383265T8+7482254pT9−607892p2T10−44105p3T11+1222p4T12+772p5T13+52p6T14−19p7T15+p8T16)2 ( 1 - 19 T + 52 T^{2} + 772 T^{3} + 1222 T^{4} - 44105 T^{5} - 607892 T^{6} + 7482254 T^{7} - 37383265 T^{8} + 7482254 p T^{9} - 607892 p^{2} T^{10} - 44105 p^{3} T^{11} + 1222 p^{4} T^{12} + 772 p^{5} T^{13} + 52 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} )^{2}
83 1+299T2+47629T4+5098189T6+383737301T8+24754599712T10+1861275246176T12+181162480129562T14+17063861270507407T16+181162480129562p2T18+1861275246176p4T20+24754599712p6T22+383737301p8T24+5098189p10T26+47629p12T28+299p14T30+p16T32 1 + 299 T^{2} + 47629 T^{4} + 5098189 T^{6} + 383737301 T^{8} + 24754599712 T^{10} + 1861275246176 T^{12} + 181162480129562 T^{14} + 17063861270507407 T^{16} + 181162480129562 p^{2} T^{18} + 1861275246176 p^{4} T^{20} + 24754599712 p^{6} T^{22} + 383737301 p^{8} T^{24} + 5098189 p^{10} T^{26} + 47629 p^{12} T^{28} + 299 p^{14} T^{30} + p^{16} T^{32}
89 (1−6T+228T2−1116T3+26613T4−1116pT5+228p2T6−6p3T7+p4T8)4 ( 1 - 6 T + 228 T^{2} - 1116 T^{3} + 26613 T^{4} - 1116 p T^{5} + 228 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{4}
97 1+412T2+78833T4+8589772T6+459983126T8−15760400260T10−6348990508023T12−840119161411750T14−86547769155085413T16−840119161411750p2T18−6348990508023p4T20−15760400260p6T22+459983126p8T24+8589772p10T26+78833p12T28+412p14T30+p16T32 1 + 412 T^{2} + 78833 T^{4} + 8589772 T^{6} + 459983126 T^{8} - 15760400260 T^{10} - 6348990508023 T^{12} - 840119161411750 T^{14} - 86547769155085413 T^{16} - 840119161411750 p^{2} T^{18} - 6348990508023 p^{4} T^{20} - 15760400260 p^{6} T^{22} + 459983126 p^{8} T^{24} + 8589772 p^{10} T^{26} + 78833 p^{12} T^{28} + 412 p^{14} T^{30} + p^{16} T^{32}
show more
show less
   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

−4.93596450467944638972142799605, −4.88531348860611381456826658394, −4.87741691774673586362181483717, −4.81495265516526242350699556021, −4.67323684807931577230985221630, −4.66639998614416985705782401760, −4.53408775954562020351447476323, −4.51009635331319408948221638034, −4.24213330381491402305082889604, −3.94035739675604629275813183242, −3.72862376075761014052961240725, −3.68106913793495170059861751868, −3.52882912728503075106484871815, −3.52100667783473373758984808484, −3.45951651040529274285335401266, −3.40210840687127325413544219627, −3.20307241903005468498017109637, −3.10293891013009170619354558803, −3.00400563869808937727519263009, −2.75212068296630253530530737040, −2.38542886778364778296386622240, −2.26510552391083776004693714983, −1.94878907718246307495164978206, −1.80567855218291239778268290432, −1.36077930952709533267681588626, 1.36077930952709533267681588626, 1.80567855218291239778268290432, 1.94878907718246307495164978206, 2.26510552391083776004693714983, 2.38542886778364778296386622240, 2.75212068296630253530530737040, 3.00400563869808937727519263009, 3.10293891013009170619354558803, 3.20307241903005468498017109637, 3.40210840687127325413544219627, 3.45951651040529274285335401266, 3.52100667783473373758984808484, 3.52882912728503075106484871815, 3.68106913793495170059861751868, 3.72862376075761014052961240725, 3.94035739675604629275813183242, 4.24213330381491402305082889604, 4.51009635331319408948221638034, 4.53408775954562020351447476323, 4.66639998614416985705782401760, 4.67323684807931577230985221630, 4.81495265516526242350699556021, 4.87741691774673586362181483717, 4.88531348860611381456826658394, 4.93596450467944638972142799605

Graph of the ZZ-function along the critical line

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