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Properties

Label 16-2175e8-1.1-c1e8-0-1
Degree 1616
Conductor 5.008×10265.008\times 10^{26}
Sign 11
Analytic cond. 8.27734×1098.27734\times 10^{9}
Root an. cond. 4.167424.16742
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  − 2·2-s − 8·3-s + 16·6-s + 2·7-s + 3·8-s + 36·9-s + 6·11-s + 6·13-s − 4·14-s − 12·17-s − 72·18-s − 16·21-s − 12·22-s − 14·23-s − 24·24-s − 12·26-s − 120·27-s + 8·29-s + 8·31-s − 4·32-s − 48·33-s + 24·34-s + 4·37-s − 48·39-s + 2·41-s + 32·42-s + 2·43-s + ⋯
L(s)  = 1  − 1.41·2-s − 4.61·3-s + 6.53·6-s + 0.755·7-s + 1.06·8-s + 12·9-s + 1.80·11-s + 1.66·13-s − 1.06·14-s − 2.91·17-s − 16.9·18-s − 3.49·21-s − 2.55·22-s − 2.91·23-s − 4.89·24-s − 2.35·26-s − 23.0·27-s + 1.48·29-s + 1.43·31-s − 0.707·32-s − 8.35·33-s + 4.11·34-s + 0.657·37-s − 7.68·39-s + 0.312·41-s + 4.93·42-s + 0.304·43-s + ⋯

Functional equation

Λ(s)=((38⋅516⋅298)s/2 ΓC(s)8 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((38⋅516⋅298)s/2 ΓC(s+1/2)8 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1616
Conductor: 38⋅516⋅2983^{8} \cdot 5^{16} \cdot 29^{8}
Sign: 11
Analytic conductor: 8.27734×1098.27734\times 10^{9}
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (16, 38⋅516⋅298, ( :[1/2]8), 1)(16,\ 3^{8} \cdot 5^{16} \cdot 29^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.0262949641.026294964
L(12)L(\frac12) ≈\approx 1.0262949641.026294964
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+T)8 ( 1 + T )^{8}
5 1 1
29 (1−T)8 ( 1 - T )^{8}
good2 1+pT+p2T2+5T3+p2T4+T6−pT7+3pT8−p2T9+p2T10+p6T12+5p5T13+p8T14+p8T15+p8T16 1 + p T + p^{2} T^{2} + 5 T^{3} + p^{2} T^{4} + T^{6} - p T^{7} + 3 p T^{8} - p^{2} T^{9} + p^{2} T^{10} + p^{6} T^{12} + 5 p^{5} T^{13} + p^{8} T^{14} + p^{8} T^{15} + p^{8} T^{16}
7 1−2T+11T2−54T3+149T4−248T5+1381T6−2678T7+4659T8−2678pT9+1381p2T10−248p3T11+149p4T12−54p5T13+11p6T14−2p7T15+p8T16 1 - 2 T + 11 T^{2} - 54 T^{3} + 149 T^{4} - 248 T^{5} + 1381 T^{6} - 2678 T^{7} + 4659 T^{8} - 2678 p T^{9} + 1381 p^{2} T^{10} - 248 p^{3} T^{11} + 149 p^{4} T^{12} - 54 p^{5} T^{13} + 11 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
11 1−6T+52T2−224T3+1277T4−4538T5+20835T6−64084T7+257338T8−64084pT9+20835p2T10−4538p3T11+1277p4T12−224p5T13+52p6T14−6p7T15+p8T16 1 - 6 T + 52 T^{2} - 224 T^{3} + 1277 T^{4} - 4538 T^{5} + 20835 T^{6} - 64084 T^{7} + 257338 T^{8} - 64084 p T^{9} + 20835 p^{2} T^{10} - 4538 p^{3} T^{11} + 1277 p^{4} T^{12} - 224 p^{5} T^{13} + 52 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
13 1−6T+43T2−136T3+837T4−2474T5+15019T6−42580T7+228905T8−42580pT9+15019p2T10−2474p3T11+837p4T12−136p5T13+43p6T14−6p7T15+p8T16 1 - 6 T + 43 T^{2} - 136 T^{3} + 837 T^{4} - 2474 T^{5} + 15019 T^{6} - 42580 T^{7} + 228905 T^{8} - 42580 p T^{9} + 15019 p^{2} T^{10} - 2474 p^{3} T^{11} + 837 p^{4} T^{12} - 136 p^{5} T^{13} + 43 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16}
17 1+12T+126T2+832T3+5153T4+25034T5+123885T6+523636T7+2323644T8+523636pT9+123885p2T10+25034p3T11+5153p4T12+832p5T13+126p6T14+12p7T15+p8T16 1 + 12 T + 126 T^{2} + 832 T^{3} + 5153 T^{4} + 25034 T^{5} + 123885 T^{6} + 523636 T^{7} + 2323644 T^{8} + 523636 p T^{9} + 123885 p^{2} T^{10} + 25034 p^{3} T^{11} + 5153 p^{4} T^{12} + 832 p^{5} T^{13} + 126 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16}
19 1+53T2−64T3+2034T4−3912T5+49091T6−125480T7+1069082T8−125480pT9+49091p2T10−3912p3T11+2034p4T12−64p5T13+53p6T14+p8T16 1 + 53 T^{2} - 64 T^{3} + 2034 T^{4} - 3912 T^{5} + 49091 T^{6} - 125480 T^{7} + 1069082 T^{8} - 125480 p T^{9} + 49091 p^{2} T^{10} - 3912 p^{3} T^{11} + 2034 p^{4} T^{12} - 64 p^{5} T^{13} + 53 p^{6} T^{14} + p^{8} T^{16}
23 1+14T+170T2+1454T3+11888T4+78870T5+496918T6+2681430T7+13842846T8+2681430pT9+496918p2T10+78870p3T11+11888p4T12+1454p5T13+170p6T14+14p7T15+p8T16 1 + 14 T + 170 T^{2} + 1454 T^{3} + 11888 T^{4} + 78870 T^{5} + 496918 T^{6} + 2681430 T^{7} + 13842846 T^{8} + 2681430 p T^{9} + 496918 p^{2} T^{10} + 78870 p^{3} T^{11} + 11888 p^{4} T^{12} + 1454 p^{5} T^{13} + 170 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16}
31 1−8T+69T2−640T3+5258T4−32016T5+234283T6−1390520T7+7909578T8−1390520pT9+234283p2T10−32016p3T11+5258p4T12−640p5T13+69p6T14−8p7T15+p8T16 1 - 8 T + 69 T^{2} - 640 T^{3} + 5258 T^{4} - 32016 T^{5} + 234283 T^{6} - 1390520 T^{7} + 7909578 T^{8} - 1390520 p T^{9} + 234283 p^{2} T^{10} - 32016 p^{3} T^{11} + 5258 p^{4} T^{12} - 640 p^{5} T^{13} + 69 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16}
37 1−4T−12T2+172T3+2372T4−1692T5−45204T6−18636T7+6125910T8−18636pT9−45204p2T10−1692p3T11+2372p4T12+172p5T13−12p6T14−4p7T15+p8T16 1 - 4 T - 12 T^{2} + 172 T^{3} + 2372 T^{4} - 1692 T^{5} - 45204 T^{6} - 18636 T^{7} + 6125910 T^{8} - 18636 p T^{9} - 45204 p^{2} T^{10} - 1692 p^{3} T^{11} + 2372 p^{4} T^{12} + 172 p^{5} T^{13} - 12 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
41 1−2T+191T2−282T3+16837T4−9752T5+959494T6+192572T7+42746654T8+192572pT9+959494p2T10−9752p3T11+16837p4T12−282p5T13+191p6T14−2p7T15+p8T16 1 - 2 T + 191 T^{2} - 282 T^{3} + 16837 T^{4} - 9752 T^{5} + 959494 T^{6} + 192572 T^{7} + 42746654 T^{8} + 192572 p T^{9} + 959494 p^{2} T^{10} - 9752 p^{3} T^{11} + 16837 p^{4} T^{12} - 282 p^{5} T^{13} + 191 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
43 1−2T+149T2+72T3+10982T4+34540T5+531475T6+3068794T7+22437618T8+3068794pT9+531475p2T10+34540p3T11+10982p4T12+72p5T13+149p6T14−2p7T15+p8T16 1 - 2 T + 149 T^{2} + 72 T^{3} + 10982 T^{4} + 34540 T^{5} + 531475 T^{6} + 3068794 T^{7} + 22437618 T^{8} + 3068794 p T^{9} + 531475 p^{2} T^{10} + 34540 p^{3} T^{11} + 10982 p^{4} T^{12} + 72 p^{5} T^{13} + 149 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
47 1+12T+4pT2+1860T3+20683T4+166890T5+1504675T6+10715322T7+1738954pT8+10715322pT9+1504675p2T10+166890p3T11+20683p4T12+1860p5T13+4p7T14+12p7T15+p8T16 1 + 12 T + 4 p T^{2} + 1860 T^{3} + 20683 T^{4} + 166890 T^{5} + 1504675 T^{6} + 10715322 T^{7} + 1738954 p T^{8} + 10715322 p T^{9} + 1504675 p^{2} T^{10} + 166890 p^{3} T^{11} + 20683 p^{4} T^{12} + 1860 p^{5} T^{13} + 4 p^{7} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16}
53 1+4T+214T2+704T3+19568T4+29592T5+1102106T6−1013740T7+54838974T8−1013740pT9+1102106p2T10+29592p3T11+19568p4T12+704p5T13+214p6T14+4p7T15+p8T16 1 + 4 T + 214 T^{2} + 704 T^{3} + 19568 T^{4} + 29592 T^{5} + 1102106 T^{6} - 1013740 T^{7} + 54838974 T^{8} - 1013740 p T^{9} + 1102106 p^{2} T^{10} + 29592 p^{3} T^{11} + 19568 p^{4} T^{12} + 704 p^{5} T^{13} + 214 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16}
59 1−18T+326T2−4506T3+51520T4−557562T5+5314762T6−46070178T7+378142430T8−46070178pT9+5314762p2T10−557562p3T11+51520p4T12−4506p5T13+326p6T14−18p7T15+p8T16 1 - 18 T + 326 T^{2} - 4506 T^{3} + 51520 T^{4} - 557562 T^{5} + 5314762 T^{6} - 46070178 T^{7} + 378142430 T^{8} - 46070178 p T^{9} + 5314762 p^{2} T^{10} - 557562 p^{3} T^{11} + 51520 p^{4} T^{12} - 4506 p^{5} T^{13} + 326 p^{6} T^{14} - 18 p^{7} T^{15} + p^{8} T^{16}
61 1−12T+277T2−3136T3+39870T4−406200T5+3782003T6−34750508T7+263133730T8−34750508pT9+3782003p2T10−406200p3T11+39870p4T12−3136p5T13+277p6T14−12p7T15+p8T16 1 - 12 T + 277 T^{2} - 3136 T^{3} + 39870 T^{4} - 406200 T^{5} + 3782003 T^{6} - 34750508 T^{7} + 263133730 T^{8} - 34750508 p T^{9} + 3782003 p^{2} T^{10} - 406200 p^{3} T^{11} + 39870 p^{4} T^{12} - 3136 p^{5} T^{13} + 277 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16}
67 1−26T+451T2−6050T3+75913T4−854712T5+8812957T6−80168870T7+682181179T8−80168870pT9+8812957p2T10−854712p3T11+75913p4T12−6050p5T13+451p6T14−26p7T15+p8T16 1 - 26 T + 451 T^{2} - 6050 T^{3} + 75913 T^{4} - 854712 T^{5} + 8812957 T^{6} - 80168870 T^{7} + 682181179 T^{8} - 80168870 p T^{9} + 8812957 p^{2} T^{10} - 854712 p^{3} T^{11} + 75913 p^{4} T^{12} - 6050 p^{5} T^{13} + 451 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16}
71 1−24T+686T2−11044T3+184584T4−2232828T5+27109682T6−257392272T7+2428813230T8−257392272pT9+27109682p2T10−2232828p3T11+184584p4T12−11044p5T13+686p6T14−24p7T15+p8T16 1 - 24 T + 686 T^{2} - 11044 T^{3} + 184584 T^{4} - 2232828 T^{5} + 27109682 T^{6} - 257392272 T^{7} + 2428813230 T^{8} - 257392272 p T^{9} + 27109682 p^{2} T^{10} - 2232828 p^{3} T^{11} + 184584 p^{4} T^{12} - 11044 p^{5} T^{13} + 686 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16}
73 1+14T+300T2+3234T3+50772T4+457014T5+5630548T6+610346pT7+477196566T8+610346p2T9+5630548p2T10+457014p3T11+50772p4T12+3234p5T13+300p6T14+14p7T15+p8T16 1 + 14 T + 300 T^{2} + 3234 T^{3} + 50772 T^{4} + 457014 T^{5} + 5630548 T^{6} + 610346 p T^{7} + 477196566 T^{8} + 610346 p^{2} T^{9} + 5630548 p^{2} T^{10} + 457014 p^{3} T^{11} + 50772 p^{4} T^{12} + 3234 p^{5} T^{13} + 300 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16}
79 1−10T+400T2−4042T3+84364T4−769890T5+11509232T6−91087970T7+1083990374T8−91087970pT9+11509232p2T10−769890p3T11+84364p4T12−4042p5T13+400p6T14−10p7T15+p8T16 1 - 10 T + 400 T^{2} - 4042 T^{3} + 84364 T^{4} - 769890 T^{5} + 11509232 T^{6} - 91087970 T^{7} + 1083990374 T^{8} - 91087970 p T^{9} + 11509232 p^{2} T^{10} - 769890 p^{3} T^{11} + 84364 p^{4} T^{12} - 4042 p^{5} T^{13} + 400 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16}
83 1+40T+916T2+15056T3+213012T4+2711520T5+31421068T6+325262680T7+3089294838T8+325262680pT9+31421068p2T10+2711520p3T11+213012p4T12+15056p5T13+916p6T14+40p7T15+p8T16 1 + 40 T + 916 T^{2} + 15056 T^{3} + 213012 T^{4} + 2711520 T^{5} + 31421068 T^{6} + 325262680 T^{7} + 3089294838 T^{8} + 325262680 p T^{9} + 31421068 p^{2} T^{10} + 2711520 p^{3} T^{11} + 213012 p^{4} T^{12} + 15056 p^{5} T^{13} + 916 p^{6} T^{14} + 40 p^{7} T^{15} + p^{8} T^{16}
89 1−34T+794T2−14230T3+218299T4−2889828T5+34814851T6−375825900T7+3720460370T8−375825900pT9+34814851p2T10−2889828p3T11+218299p4T12−14230p5T13+794p6T14−34p7T15+p8T16 1 - 34 T + 794 T^{2} - 14230 T^{3} + 218299 T^{4} - 2889828 T^{5} + 34814851 T^{6} - 375825900 T^{7} + 3720460370 T^{8} - 375825900 p T^{9} + 34814851 p^{2} T^{10} - 2889828 p^{3} T^{11} + 218299 p^{4} T^{12} - 14230 p^{5} T^{13} + 794 p^{6} T^{14} - 34 p^{7} T^{15} + p^{8} T^{16}
97 1−30T+893T2−16084T3+286266T4−3832848T5+51219155T6−551902430T7+5992632938T8−551902430pT9+51219155p2T10−3832848p3T11+286266p4T12−16084p5T13+893p6T14−30p7T15+p8T16 1 - 30 T + 893 T^{2} - 16084 T^{3} + 286266 T^{4} - 3832848 T^{5} + 51219155 T^{6} - 551902430 T^{7} + 5992632938 T^{8} - 551902430 p T^{9} + 51219155 p^{2} T^{10} - 3832848 p^{3} T^{11} + 286266 p^{4} T^{12} - 16084 p^{5} T^{13} + 893 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16}
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   L(s)=∏p ∏j=116(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.90458636026905220803120738912, −3.67450135568525205193518487457, −3.60937052124579437279199746164, −3.58670327736621200527163270118, −3.52534545507026369945491233549, −3.36305665104416248127961615640, −3.09276178305079768419663398661, −2.92319217425504110822603056912, −2.65670989075033562603567888800, −2.60968081231717799704858326995, −2.24972894538789046472460918091, −2.22887168477300029850118123728, −2.22533892996369667379036589032, −1.80901281345292221085470937463, −1.79080821953127293385960464746, −1.69678880257147556323904755680, −1.59030735987841178139386653254, −1.47834099910361017075999504871, −1.14884792375178383344801016190, −0.983878420232591295657335447188, −0.807133543343497993372000165724, −0.59146369806927440250452264530, −0.56006473139472308493909671056, −0.47509649146166903336571070911, −0.36888289924507259664087467196, 0.36888289924507259664087467196, 0.47509649146166903336571070911, 0.56006473139472308493909671056, 0.59146369806927440250452264530, 0.807133543343497993372000165724, 0.983878420232591295657335447188, 1.14884792375178383344801016190, 1.47834099910361017075999504871, 1.59030735987841178139386653254, 1.69678880257147556323904755680, 1.79080821953127293385960464746, 1.80901281345292221085470937463, 2.22533892996369667379036589032, 2.22887168477300029850118123728, 2.24972894538789046472460918091, 2.60968081231717799704858326995, 2.65670989075033562603567888800, 2.92319217425504110822603056912, 3.09276178305079768419663398661, 3.36305665104416248127961615640, 3.52534545507026369945491233549, 3.58670327736621200527163270118, 3.60937052124579437279199746164, 3.67450135568525205193518487457, 3.90458636026905220803120738912

Graph of the ZZ-function along the critical line

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