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Properties

Label 18-3971e9-1.1-c1e9-0-0
Degree 1818
Conductor 2.455×10322.455\times 10^{32}
Sign 11
Analytic cond. 3.24035×10133.24035\times 10^{13}
Root an. cond. 5.631035.63103
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  + 2-s − 4·4-s − 3·7-s − 5·8-s − 8·9-s − 9·11-s + 11·13-s − 3·14-s + 6·16-s − 6·17-s − 8·18-s − 9·22-s + 23-s − 19·25-s + 11·26-s − 27-s + 12·28-s + 13·29-s + 16·31-s + 8·32-s − 6·34-s + 32·36-s + 18·37-s + 4·41-s − 43-s + 36·44-s + 46-s + ⋯
L(s)  = 1  + 0.707·2-s − 2·4-s − 1.13·7-s − 1.76·8-s − 8/3·9-s − 2.71·11-s + 3.05·13-s − 0.801·14-s + 3/2·16-s − 1.45·17-s − 1.88·18-s − 1.91·22-s + 0.208·23-s − 3.79·25-s + 2.15·26-s − 0.192·27-s + 2.26·28-s + 2.41·29-s + 2.87·31-s + 1.41·32-s − 1.02·34-s + 16/3·36-s + 2.95·37-s + 0.624·41-s − 0.152·43-s + 5.42·44-s + 0.147·46-s + ⋯

Functional equation

Λ(s)=((119⋅1918)s/2 ΓC(s)9 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{9} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((119⋅1918)s/2 ΓC(s+1/2)9 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{9} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{9} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1818
Conductor: 119⋅191811^{9} \cdot 19^{18}
Sign: 11
Analytic conductor: 3.24035×10133.24035\times 10^{13}
Root analytic conductor: 5.631035.63103
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (18, 119⋅1918, ( :[1/2]9), 1)(18,\ 11^{9} \cdot 19^{18} ,\ ( \ : [1/2]^{9} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 5.6363817525.636381752
L(12)L(\frac12) ≈\approx 5.6363817525.636381752
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)
bad11 (1+T)9 ( 1 + T )^{9}
19 1 1
good2 1−T+5T2−p2T3+13T4−3pT5+5p2T6−pT7+23T8+11T9+23pT10−p3T11+5p5T12−3p5T13+13p5T14−p8T15+5p7T16−p8T17+p9T18 1 - T + 5 T^{2} - p^{2} T^{3} + 13 T^{4} - 3 p T^{5} + 5 p^{2} T^{6} - p T^{7} + 23 T^{8} + 11 T^{9} + 23 p T^{10} - p^{3} T^{11} + 5 p^{5} T^{12} - 3 p^{5} T^{13} + 13 p^{5} T^{14} - p^{8} T^{15} + 5 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18}
3 1+8T2+T3+35T4+103T6−46T7+283T8−236T9+283pT10−46p2T11+103p3T12+35p5T14+p6T15+8p7T16+p9T18 1 + 8 T^{2} + T^{3} + 35 T^{4} + 103 T^{6} - 46 T^{7} + 283 T^{8} - 236 T^{9} + 283 p T^{10} - 46 p^{2} T^{11} + 103 p^{3} T^{12} + 35 p^{5} T^{14} + p^{6} T^{15} + 8 p^{7} T^{16} + p^{9} T^{18}
5 1+19T2+7T3+201T4+81T5+1589T6+558T7+9788T8+3193T9+9788pT10+558p2T11+1589p3T12+81p4T13+201p5T14+7p6T15+19p7T16+p9T18 1 + 19 T^{2} + 7 T^{3} + 201 T^{4} + 81 T^{5} + 1589 T^{6} + 558 T^{7} + 9788 T^{8} + 3193 T^{9} + 9788 p T^{10} + 558 p^{2} T^{11} + 1589 p^{3} T^{12} + 81 p^{4} T^{13} + 201 p^{5} T^{14} + 7 p^{6} T^{15} + 19 p^{7} T^{16} + p^{9} T^{18}
7 1+3T+39T2+104T3+745T4+1780T5+9367T6+19805T7+86060T8+159832T9+86060pT10+19805p2T11+9367p3T12+1780p4T13+745p5T14+104p6T15+39p7T16+3p8T17+p9T18 1 + 3 T + 39 T^{2} + 104 T^{3} + 745 T^{4} + 1780 T^{5} + 9367 T^{6} + 19805 T^{7} + 86060 T^{8} + 159832 T^{9} + 86060 p T^{10} + 19805 p^{2} T^{11} + 9367 p^{3} T^{12} + 1780 p^{4} T^{13} + 745 p^{5} T^{14} + 104 p^{6} T^{15} + 39 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18}
13 1−11T+119T2−813T3+5400T4−27993T5+10943pT6−605116T7+2545975T8−9210564T9+2545975pT10−605116p2T11+10943p4T12−27993p4T13+5400p5T14−813p6T15+119p7T16−11p8T17+p9T18 1 - 11 T + 119 T^{2} - 813 T^{3} + 5400 T^{4} - 27993 T^{5} + 10943 p T^{6} - 605116 T^{7} + 2545975 T^{8} - 9210564 T^{9} + 2545975 p T^{10} - 605116 p^{2} T^{11} + 10943 p^{4} T^{12} - 27993 p^{4} T^{13} + 5400 p^{5} T^{14} - 813 p^{6} T^{15} + 119 p^{7} T^{16} - 11 p^{8} T^{17} + p^{9} T^{18}
17 1+6T+111T2+617T3+5790T4+30432T5+191081T6+929784T7+4435197T8+19079666T9+4435197pT10+929784p2T11+191081p3T12+30432p4T13+5790p5T14+617p6T15+111p7T16+6p8T17+p9T18 1 + 6 T + 111 T^{2} + 617 T^{3} + 5790 T^{4} + 30432 T^{5} + 191081 T^{6} + 929784 T^{7} + 4435197 T^{8} + 19079666 T^{9} + 4435197 p T^{10} + 929784 p^{2} T^{11} + 191081 p^{3} T^{12} + 30432 p^{4} T^{13} + 5790 p^{5} T^{14} + 617 p^{6} T^{15} + 111 p^{7} T^{16} + 6 p^{8} T^{17} + p^{9} T^{18}
23 1−T+147T2−70T3+10085T4+248T5+433001T6+159707T7+13203986T8+5940710T9+13203986pT10+159707p2T11+433001p3T12+248p4T13+10085p5T14−70p6T15+147p7T16−p8T17+p9T18 1 - T + 147 T^{2} - 70 T^{3} + 10085 T^{4} + 248 T^{5} + 433001 T^{6} + 159707 T^{7} + 13203986 T^{8} + 5940710 T^{9} + 13203986 p T^{10} + 159707 p^{2} T^{11} + 433001 p^{3} T^{12} + 248 p^{4} T^{13} + 10085 p^{5} T^{14} - 70 p^{6} T^{15} + 147 p^{7} T^{16} - p^{8} T^{17} + p^{9} T^{18}
29 1−13T+223T2−1853T3+19394T4−126757T5+1052735T6−5884726T7+41347885T8−199424468T9+41347885pT10−5884726p2T11+1052735p3T12−126757p4T13+19394p5T14−1853p6T15+223p7T16−13p8T17+p9T18 1 - 13 T + 223 T^{2} - 1853 T^{3} + 19394 T^{4} - 126757 T^{5} + 1052735 T^{6} - 5884726 T^{7} + 41347885 T^{8} - 199424468 T^{9} + 41347885 p T^{10} - 5884726 p^{2} T^{11} + 1052735 p^{3} T^{12} - 126757 p^{4} T^{13} + 19394 p^{5} T^{14} - 1853 p^{6} T^{15} + 223 p^{7} T^{16} - 13 p^{8} T^{17} + p^{9} T^{18}
31 1−16T+204T2−1637T3+12380T4−74671T5+16592pT6−3228381T7+22145447T8−124293758T9+22145447pT10−3228381p2T11+16592p4T12−74671p4T13+12380p5T14−1637p6T15+204p7T16−16p8T17+p9T18 1 - 16 T + 204 T^{2} - 1637 T^{3} + 12380 T^{4} - 74671 T^{5} + 16592 p T^{6} - 3228381 T^{7} + 22145447 T^{8} - 124293758 T^{9} + 22145447 p T^{10} - 3228381 p^{2} T^{11} + 16592 p^{4} T^{12} - 74671 p^{4} T^{13} + 12380 p^{5} T^{14} - 1637 p^{6} T^{15} + 204 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18}
37 1−18T+255T2−2371T3+19420T4−126102T5+817729T6−4747378T7+30516247T8−177678966T9+30516247pT10−4747378p2T11+817729p3T12−126102p4T13+19420p5T14−2371p6T15+255p7T16−18p8T17+p9T18 1 - 18 T + 255 T^{2} - 2371 T^{3} + 19420 T^{4} - 126102 T^{5} + 817729 T^{6} - 4747378 T^{7} + 30516247 T^{8} - 177678966 T^{9} + 30516247 p T^{10} - 4747378 p^{2} T^{11} + 817729 p^{3} T^{12} - 126102 p^{4} T^{13} + 19420 p^{5} T^{14} - 2371 p^{6} T^{15} + 255 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18}
41 1−4T+102T2−466T3+7319T4−40452T5+399061T6−2450030T7+19484933T8−117806128T9+19484933pT10−2450030p2T11+399061p3T12−40452p4T13+7319p5T14−466p6T15+102p7T16−4p8T17+p9T18 1 - 4 T + 102 T^{2} - 466 T^{3} + 7319 T^{4} - 40452 T^{5} + 399061 T^{6} - 2450030 T^{7} + 19484933 T^{8} - 117806128 T^{9} + 19484933 p T^{10} - 2450030 p^{2} T^{11} + 399061 p^{3} T^{12} - 40452 p^{4} T^{13} + 7319 p^{5} T^{14} - 466 p^{6} T^{15} + 102 p^{7} T^{16} - 4 p^{8} T^{17} + p^{9} T^{18}
43 1+T+245T2+336T3+29619T4+49988T5+54415pT6+4238487T7+133607998T8+226251824T9+133607998pT10+4238487p2T11+54415p4T12+49988p4T13+29619p5T14+336p6T15+245p7T16+p8T17+p9T18 1 + T + 245 T^{2} + 336 T^{3} + 29619 T^{4} + 49988 T^{5} + 54415 p T^{6} + 4238487 T^{7} + 133607998 T^{8} + 226251824 T^{9} + 133607998 p T^{10} + 4238487 p^{2} T^{11} + 54415 p^{4} T^{12} + 49988 p^{4} T^{13} + 29619 p^{5} T^{14} + 336 p^{6} T^{15} + 245 p^{7} T^{16} + p^{8} T^{17} + p^{9} T^{18}
47 1−14T+389T2−4450T3+67899T4−643673T5+7043097T6−55779851T7+481927342T8−3186952630T9+481927342pT10−55779851p2T11+7043097p3T12−643673p4T13+67899p5T14−4450p6T15+389p7T16−14p8T17+p9T18 1 - 14 T + 389 T^{2} - 4450 T^{3} + 67899 T^{4} - 643673 T^{5} + 7043097 T^{6} - 55779851 T^{7} + 481927342 T^{8} - 3186952630 T^{9} + 481927342 p T^{10} - 55779851 p^{2} T^{11} + 7043097 p^{3} T^{12} - 643673 p^{4} T^{13} + 67899 p^{5} T^{14} - 4450 p^{6} T^{15} + 389 p^{7} T^{16} - 14 p^{8} T^{17} + p^{9} T^{18}
53 1+4T+262T2+992T3+29948T4+90003T5+2045897T6+3553202T7+106752274T8+103234313T9+106752274pT10+3553202p2T11+2045897p3T12+90003p4T13+29948p5T14+992p6T15+262p7T16+4p8T17+p9T18 1 + 4 T + 262 T^{2} + 992 T^{3} + 29948 T^{4} + 90003 T^{5} + 2045897 T^{6} + 3553202 T^{7} + 106752274 T^{8} + 103234313 T^{9} + 106752274 p T^{10} + 3553202 p^{2} T^{11} + 2045897 p^{3} T^{12} + 90003 p^{4} T^{13} + 29948 p^{5} T^{14} + 992 p^{6} T^{15} + 262 p^{7} T^{16} + 4 p^{8} T^{17} + p^{9} T^{18}
59 1−31T+733T2−12305T3+178108T4−2169823T5+23798005T6−231234820T7+2056073739T8−16473750500T9+2056073739pT10−231234820p2T11+23798005p3T12−2169823p4T13+178108p5T14−12305p6T15+733p7T16−31p8T17+p9T18 1 - 31 T + 733 T^{2} - 12305 T^{3} + 178108 T^{4} - 2169823 T^{5} + 23798005 T^{6} - 231234820 T^{7} + 2056073739 T^{8} - 16473750500 T^{9} + 2056073739 p T^{10} - 231234820 p^{2} T^{11} + 23798005 p^{3} T^{12} - 2169823 p^{4} T^{13} + 178108 p^{5} T^{14} - 12305 p^{6} T^{15} + 733 p^{7} T^{16} - 31 p^{8} T^{17} + p^{9} T^{18}
61 1+3T+333T2+810T3+54985T4+106406T5+5915573T6+9104805T7+466642044T8+607593492T9+466642044pT10+9104805p2T11+5915573p3T12+106406p4T13+54985p5T14+810p6T15+333p7T16+3p8T17+p9T18 1 + 3 T + 333 T^{2} + 810 T^{3} + 54985 T^{4} + 106406 T^{5} + 5915573 T^{6} + 9104805 T^{7} + 466642044 T^{8} + 607593492 T^{9} + 466642044 p T^{10} + 9104805 p^{2} T^{11} + 5915573 p^{3} T^{12} + 106406 p^{4} T^{13} + 54985 p^{5} T^{14} + 810 p^{6} T^{15} + 333 p^{7} T^{16} + 3 p^{8} T^{17} + p^{9} T^{18}
67 1+2T+173T2−900T3+16941T4−154631T5+1909993T6−13954719T7+169271824T8−1036918952T9+169271824pT10−13954719p2T11+1909993p3T12−154631p4T13+16941p5T14−900p6T15+173p7T16+2p8T17+p9T18 1 + 2 T + 173 T^{2} - 900 T^{3} + 16941 T^{4} - 154631 T^{5} + 1909993 T^{6} - 13954719 T^{7} + 169271824 T^{8} - 1036918952 T^{9} + 169271824 p T^{10} - 13954719 p^{2} T^{11} + 1909993 p^{3} T^{12} - 154631 p^{4} T^{13} + 16941 p^{5} T^{14} - 900 p^{6} T^{15} + 173 p^{7} T^{16} + 2 p^{8} T^{17} + p^{9} T^{18}
71 1+12T+482T2+4271T3+101775T4+706390T5+13179487T6+75020186T7+1217547613T8+5975256326T9+1217547613pT10+75020186p2T11+13179487p3T12+706390p4T13+101775p5T14+4271p6T15+482p7T16+12p8T17+p9T18 1 + 12 T + 482 T^{2} + 4271 T^{3} + 101775 T^{4} + 706390 T^{5} + 13179487 T^{6} + 75020186 T^{7} + 1217547613 T^{8} + 5975256326 T^{9} + 1217547613 p T^{10} + 75020186 p^{2} T^{11} + 13179487 p^{3} T^{12} + 706390 p^{4} T^{13} + 101775 p^{5} T^{14} + 4271 p^{6} T^{15} + 482 p^{7} T^{16} + 12 p^{8} T^{17} + p^{9} T^{18}
73 1−26T+735T2−12394T3+209699T4−2688291T5+34149005T6−354861739T7+3639401356T8−31323905962T9+3639401356pT10−354861739p2T11+34149005p3T12−2688291p4T13+209699p5T14−12394p6T15+735p7T16−26p8T17+p9T18 1 - 26 T + 735 T^{2} - 12394 T^{3} + 209699 T^{4} - 2688291 T^{5} + 34149005 T^{6} - 354861739 T^{7} + 3639401356 T^{8} - 31323905962 T^{9} + 3639401356 p T^{10} - 354861739 p^{2} T^{11} + 34149005 p^{3} T^{12} - 2688291 p^{4} T^{13} + 209699 p^{5} T^{14} - 12394 p^{6} T^{15} + 735 p^{7} T^{16} - 26 p^{8} T^{17} + p^{9} T^{18}
79 1−31T+742T2−14342T3+229347T4−3223040T5+40513046T6−455576998T7+4654775821T8−43478831993T9+4654775821pT10−455576998p2T11+40513046p3T12−3223040p4T13+229347p5T14−14342p6T15+742p7T16−31p8T17+p9T18 1 - 31 T + 742 T^{2} - 14342 T^{3} + 229347 T^{4} - 3223040 T^{5} + 40513046 T^{6} - 455576998 T^{7} + 4654775821 T^{8} - 43478831993 T^{9} + 4654775821 p T^{10} - 455576998 p^{2} T^{11} + 40513046 p^{3} T^{12} - 3223040 p^{4} T^{13} + 229347 p^{5} T^{14} - 14342 p^{6} T^{15} + 742 p^{7} T^{16} - 31 p^{8} T^{17} + p^{9} T^{18}
83 1−18T+592T2−8721T3+156875T4−1984688T5+25392575T6−281639548T7+2859446887T8−27675516522T9+2859446887pT10−281639548p2T11+25392575p3T12−1984688p4T13+156875p5T14−8721p6T15+592p7T16−18p8T17+p9T18 1 - 18 T + 592 T^{2} - 8721 T^{3} + 156875 T^{4} - 1984688 T^{5} + 25392575 T^{6} - 281639548 T^{7} + 2859446887 T^{8} - 27675516522 T^{9} + 2859446887 p T^{10} - 281639548 p^{2} T^{11} + 25392575 p^{3} T^{12} - 1984688 p^{4} T^{13} + 156875 p^{5} T^{14} - 8721 p^{6} T^{15} + 592 p^{7} T^{16} - 18 p^{8} T^{17} + p^{9} T^{18}
89 1+11T+549T2+4995T3+1502pT4+981629T5+19768773T6+117305546T7+23914739pT8+10991786538T9+23914739p2T10+117305546p2T11+19768773p3T12+981629p4T13+1502p6T14+4995p6T15+549p7T16+11p8T17+p9T18 1 + 11 T + 549 T^{2} + 4995 T^{3} + 1502 p T^{4} + 981629 T^{5} + 19768773 T^{6} + 117305546 T^{7} + 23914739 p T^{8} + 10991786538 T^{9} + 23914739 p^{2} T^{10} + 117305546 p^{2} T^{11} + 19768773 p^{3} T^{12} + 981629 p^{4} T^{13} + 1502 p^{6} T^{14} + 4995 p^{6} T^{15} + 549 p^{7} T^{16} + 11 p^{8} T^{17} + p^{9} T^{18}
97 1−16T+302T2−2744T3+37192T4−307461T5+5192125T6−42248018T7+570256950T8−4194757543T9+570256950pT10−42248018p2T11+5192125p3T12−307461p4T13+37192p5T14−2744p6T15+302p7T16−16p8T17+p9T18 1 - 16 T + 302 T^{2} - 2744 T^{3} + 37192 T^{4} - 307461 T^{5} + 5192125 T^{6} - 42248018 T^{7} + 570256950 T^{8} - 4194757543 T^{9} + 570256950 p T^{10} - 42248018 p^{2} T^{11} + 5192125 p^{3} T^{12} - 307461 p^{4} T^{13} + 37192 p^{5} T^{14} - 2744 p^{6} T^{15} + 302 p^{7} T^{16} - 16 p^{8} T^{17} + p^{9} T^{18}
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   L(s)=∏p ∏j=118(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.13889567592382093711666147100, −3.07932697147533834143305337925, −3.07732368638354055432364002204, −2.97040640940708571810762122708, −2.94261263769254497762922203794, −2.73106870086836540342027545088, −2.47162303233855025253627227893, −2.40921043000739861268056007269, −2.37617446364266621506029660228, −2.26449448988682815268792259238, −2.22640354267102376513955543162, −2.15596250493065475270497708474, −1.99492257291340873635942229756, −1.73598120307268340635087218015, −1.65029667241330541301024418459, −1.64184389458348514774280727959, −1.20221735992101812742114563677, −1.07406684116710787277961086867, −1.05696608622015937956569481003, −0.797850546832835060022784538220, −0.64901126752227560035742641770, −0.44026495043847370435503332121, −0.43589813136329832599549602497, −0.42287732364286712625452112062, −0.28706210617433026823864249467, 0.28706210617433026823864249467, 0.42287732364286712625452112062, 0.43589813136329832599549602497, 0.44026495043847370435503332121, 0.64901126752227560035742641770, 0.797850546832835060022784538220, 1.05696608622015937956569481003, 1.07406684116710787277961086867, 1.20221735992101812742114563677, 1.64184389458348514774280727959, 1.65029667241330541301024418459, 1.73598120307268340635087218015, 1.99492257291340873635942229756, 2.15596250493065475270497708474, 2.22640354267102376513955543162, 2.26449448988682815268792259238, 2.37617446364266621506029660228, 2.40921043000739861268056007269, 2.47162303233855025253627227893, 2.73106870086836540342027545088, 2.94261263769254497762922203794, 2.97040640940708571810762122708, 3.07732368638354055432364002204, 3.07932697147533834143305337925, 3.13889567592382093711666147100

Graph of the ZZ-function along the critical line

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