Properties

Label 10-208e5-1.1-c9e5-0-1
Degree 1010
Conductor 389328928768389328928768
Sign 1-1
Analytic cond. 1.41092×10101.41092\times 10^{10}
Root an. cond. 10.350210.3502
Motivic weight 99
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 55

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 161·3-s + 1.80e3·5-s − 1.00e4·7-s − 5.71e3·9-s − 1.21e5·11-s + 1.42e5·13-s − 2.90e5·15-s − 4.95e5·17-s + 8.40e5·19-s + 1.62e6·21-s + 5.92e5·23-s − 2.42e6·25-s + 1.08e6·27-s + 1.06e7·29-s − 1.28e7·31-s + 1.96e7·33-s − 1.82e7·35-s + 7.17e6·37-s − 2.29e7·39-s + 9.29e6·41-s − 1.28e7·43-s − 1.03e7·45-s − 4.33e7·47-s − 3.72e7·49-s + 7.98e7·51-s + 9.32e7·53-s − 2.19e8·55-s + ⋯
L(s)  = 1  − 1.14·3-s + 1.29·5-s − 1.58·7-s − 0.290·9-s − 2.50·11-s + 1.38·13-s − 1.48·15-s − 1.43·17-s + 1.48·19-s + 1.82·21-s + 0.441·23-s − 1.24·25-s + 0.392·27-s + 2.80·29-s − 2.50·31-s + 2.87·33-s − 2.05·35-s + 0.629·37-s − 1.59·39-s + 0.513·41-s − 0.572·43-s − 0.374·45-s − 1.29·47-s − 0.923·49-s + 1.65·51-s + 1.62·53-s − 3.23·55-s + ⋯

Functional equation

Λ(s)=((220135)s/2ΓC(s)5L(s)=(Λ(10s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}
Λ(s)=((220135)s/2ΓC(s+9/2)5L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1010
Conductor: 2201352^{20} \cdot 13^{5}
Sign: 1-1
Analytic conductor: 1.41092×10101.41092\times 10^{10}
Root analytic conductor: 10.350210.3502
Motivic weight: 99
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 55
Selberg data: (10, 220135, ( :9/2,9/2,9/2,9/2,9/2), 1)(10,\ 2^{20} \cdot 13^{5} ,\ ( \ : 9/2, 9/2, 9/2, 9/2, 9/2 ),\ -1 )

Particular Values

L(5)L(5) == 00
L(12)L(\frac12) == 00
L(112)L(\frac{11}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
13C1C_1 (1p4T)5 ( 1 - p^{4} T )^{5}
good3C2S5C_2 \wr S_5 1+161T+10546pT2+60851p4T3+37844987p3T4+1621483792p4T5+37844987p12T6+60851p22T7+10546p28T8+161p36T9+p45T10 1 + 161 T + 10546 p T^{2} + 60851 p^{4} T^{3} + 37844987 p^{3} T^{4} + 1621483792 p^{4} T^{5} + 37844987 p^{12} T^{6} + 60851 p^{22} T^{7} + 10546 p^{28} T^{8} + 161 p^{36} T^{9} + p^{45} T^{10}
5C2S5C_2 \wr S_5 11803T+5673036T22122118037pT3+154127491547p3T4217433194071432p3T5+154127491547p12T62122118037p19T7+5673036p27T81803p36T9+p45T10 1 - 1803 T + 5673036 T^{2} - 2122118037 p T^{3} + 154127491547 p^{3} T^{4} - 217433194071432 p^{3} T^{5} + 154127491547 p^{12} T^{6} - 2122118037 p^{19} T^{7} + 5673036 p^{27} T^{8} - 1803 p^{36} T^{9} + p^{45} T^{10}
7C2S5C_2 \wr S_5 1+10099T+139254174T2+122453891247pT3+166521050648013p2T4+122374113130618344p3T5+166521050648013p11T6+122453891247p19T7+139254174p27T8+10099p36T9+p45T10 1 + 10099 T + 139254174 T^{2} + 122453891247 p T^{3} + 166521050648013 p^{2} T^{4} + 122374113130618344 p^{3} T^{5} + 166521050648013 p^{11} T^{6} + 122453891247 p^{19} T^{7} + 139254174 p^{27} T^{8} + 10099 p^{36} T^{9} + p^{45} T^{10}
11C2S5C_2 \wr S_5 1+121746T+14269408587T2+1047597531024144T3+70977489220411684558T4+ 1 + 121746 T + 14269408587 T^{2} + 1047597531024144 T^{3} + 70977489220411684558 T^{4} + 36 ⁣ ⁣5236\!\cdots\!52T5+70977489220411684558p9T6+1047597531024144p18T7+14269408587p27T8+121746p36T9+p45T10 T^{5} + 70977489220411684558 p^{9} T^{6} + 1047597531024144 p^{18} T^{7} + 14269408587 p^{27} T^{8} + 121746 p^{36} T^{9} + p^{45} T^{10}
17C2S5C_2 \wr S_5 1+29157pT+303748065600T2+136238774144131095T3+ 1 + 29157 p T + 303748065600 T^{2} + 136238774144131095 T^{3} + 32 ⁣ ⁣8332\!\cdots\!83pT4+ p T^{4} + 62 ⁣ ⁣6462\!\cdots\!64p2T5+ p^{2} T^{5} + 32 ⁣ ⁣8332\!\cdots\!83p10T6+136238774144131095p18T7+303748065600p27T8+29157p37T9+p45T10 p^{10} T^{6} + 136238774144131095 p^{18} T^{7} + 303748065600 p^{27} T^{8} + 29157 p^{37} T^{9} + p^{45} T^{10}
19C2S5C_2 \wr S_5 1840738T+1352453971283T2953145467206471696T3+ 1 - 840738 T + 1352453971283 T^{2} - 953145467206471696 T^{3} + 83 ⁣ ⁣3483\!\cdots\!34T4 T^{4} - 43 ⁣ ⁣5243\!\cdots\!52T5+ T^{5} + 83 ⁣ ⁣3483\!\cdots\!34p9T6953145467206471696p18T7+1352453971283p27T8840738p36T9+p45T10 p^{9} T^{6} - 953145467206471696 p^{18} T^{7} + 1352453971283 p^{27} T^{8} - 840738 p^{36} T^{9} + p^{45} T^{10}
23C2S5C_2 \wr S_5 1592152T+5512736304883T2+937525593298362720T3+ 1 - 592152 T + 5512736304883 T^{2} + 937525593298362720 T^{3} + 10 ⁣ ⁣9410\!\cdots\!94T4+ T^{4} + 33 ⁣ ⁣7633\!\cdots\!76pT5+ p T^{5} + 10 ⁣ ⁣9410\!\cdots\!94p9T6+937525593298362720p18T7+5512736304883p27T8592152p36T9+p45T10 p^{9} T^{6} + 937525593298362720 p^{18} T^{7} + 5512736304883 p^{27} T^{8} - 592152 p^{36} T^{9} + p^{45} T^{10}
29C2S5C_2 \wr S_5 110678182T+88796663366937T2 1 - 10678182 T + 88796663366937 T^{2} - 54 ⁣ ⁣5654\!\cdots\!56T3+ T^{3} + 28 ⁣ ⁣7828\!\cdots\!78T4 T^{4} - 11 ⁣ ⁣4411\!\cdots\!44T5+ T^{5} + 28 ⁣ ⁣7828\!\cdots\!78p9T6 p^{9} T^{6} - 54 ⁣ ⁣5654\!\cdots\!56p18T7+88796663366937p27T810678182p36T9+p45T10 p^{18} T^{7} + 88796663366937 p^{27} T^{8} - 10678182 p^{36} T^{9} + p^{45} T^{10}
31C2S5C_2 \wr S_5 1+12885296T+164536395807867T2+ 1 + 12885296 T + 164536395807867 T^{2} + 12 ⁣ ⁣2412\!\cdots\!24T3+ T^{3} + 92 ⁣ ⁣7892\!\cdots\!78T4+ T^{4} + 48 ⁣ ⁣7248\!\cdots\!72T5+ T^{5} + 92 ⁣ ⁣7892\!\cdots\!78p9T6+ p^{9} T^{6} + 12 ⁣ ⁣2412\!\cdots\!24p18T7+164536395807867p27T8+12885296p36T9+p45T10 p^{18} T^{7} + 164536395807867 p^{27} T^{8} + 12885296 p^{36} T^{9} + p^{45} T^{10}
37C2S5C_2 \wr S_5 17171823T+354892872282468T2 1 - 7171823 T + 354892872282468 T^{2} - 10 ⁣ ⁣1310\!\cdots\!13T3+ T^{3} + 60 ⁣ ⁣7960\!\cdots\!79T4 T^{4} - 91 ⁣ ⁣1691\!\cdots\!16T5+ T^{5} + 60 ⁣ ⁣7960\!\cdots\!79p9T6 p^{9} T^{6} - 10 ⁣ ⁣1310\!\cdots\!13p18T7+354892872282468p27T87171823p36T9+p45T10 p^{18} T^{7} + 354892872282468 p^{27} T^{8} - 7171823 p^{36} T^{9} + p^{45} T^{10}
41C2S5C_2 \wr S_5 19294012T+571994210722321T2 1 - 9294012 T + 571994210722321 T^{2} - 89 ⁣ ⁣6089\!\cdots\!60T3+ T^{3} + 16 ⁣ ⁣0216\!\cdots\!02T4 T^{4} - 34 ⁣ ⁣0434\!\cdots\!04T5+ T^{5} + 16 ⁣ ⁣0216\!\cdots\!02p9T6 p^{9} T^{6} - 89 ⁣ ⁣6089\!\cdots\!60p18T7+571994210722321p27T89294012p36T9+p45T10 p^{18} T^{7} + 571994210722321 p^{27} T^{8} - 9294012 p^{36} T^{9} + p^{45} T^{10}
43C2S5C_2 \wr S_5 1+12831975T+322694905923638T2+ 1 + 12831975 T + 322694905923638 T^{2} + 26 ⁣ ⁣8926\!\cdots\!89T3+ T^{3} + 62 ⁣ ⁣7762\!\cdots\!77T4+ T^{4} + 43 ⁣ ⁣9643\!\cdots\!96T5+ T^{5} + 62 ⁣ ⁣7762\!\cdots\!77p9T6+ p^{9} T^{6} + 26 ⁣ ⁣8926\!\cdots\!89p18T7+322694905923638p27T8+12831975p36T9+p45T10 p^{18} T^{7} + 322694905923638 p^{27} T^{8} + 12831975 p^{36} T^{9} + p^{45} T^{10}
47C2S5C_2 \wr S_5 1+43354215T+2527109677831630T2+ 1 + 43354215 T + 2527109677831630 T^{2} + 75 ⁣ ⁣9375\!\cdots\!93T3+ T^{3} + 33 ⁣ ⁣4133\!\cdots\!41T4+ T^{4} + 71 ⁣ ⁣5671\!\cdots\!56T5+ T^{5} + 33 ⁣ ⁣4133\!\cdots\!41p9T6+ p^{9} T^{6} + 75 ⁣ ⁣9375\!\cdots\!93p18T7+2527109677831630p27T8+43354215p36T9+p45T10 p^{18} T^{7} + 2527109677831630 p^{27} T^{8} + 43354215 p^{36} T^{9} + p^{45} T^{10}
53C2S5C_2 \wr S_5 193231780T+14353270896409645T2 1 - 93231780 T + 14353270896409645 T^{2} - 11 ⁣ ⁣9611\!\cdots\!96T3+ T^{3} + 88 ⁣ ⁣0688\!\cdots\!06T4 T^{4} - 52 ⁣ ⁣0852\!\cdots\!08T5+ T^{5} + 88 ⁣ ⁣0688\!\cdots\!06p9T6 p^{9} T^{6} - 11 ⁣ ⁣9611\!\cdots\!96p18T7+14353270896409645p27T893231780p36T9+p45T10 p^{18} T^{7} + 14353270896409645 p^{27} T^{8} - 93231780 p^{36} T^{9} + p^{45} T^{10}
59C2S5C_2 \wr S_5 1+246496182T+33223507046890507T2+ 1 + 246496182 T + 33223507046890507 T^{2} + 26 ⁣ ⁣6026\!\cdots\!60T3+ T^{3} + 17 ⁣ ⁣5017\!\cdots\!50T4+ T^{4} + 12 ⁣ ⁣3612\!\cdots\!36T5+ T^{5} + 17 ⁣ ⁣5017\!\cdots\!50p9T6+ p^{9} T^{6} + 26 ⁣ ⁣6026\!\cdots\!60p18T7+33223507046890507p27T8+246496182p36T9+p45T10 p^{18} T^{7} + 33223507046890507 p^{27} T^{8} + 246496182 p^{36} T^{9} + p^{45} T^{10}
61C2S5C_2 \wr S_5 1+132232612T+51932556467342037T2+ 1 + 132232612 T + 51932556467342037 T^{2} + 51 ⁣ ⁣1251\!\cdots\!12T3+ T^{3} + 11 ⁣ ⁣7411\!\cdots\!74T4+ T^{4} + 84 ⁣ ⁣4884\!\cdots\!48T5+ T^{5} + 11 ⁣ ⁣7411\!\cdots\!74p9T6+ p^{9} T^{6} + 51 ⁣ ⁣1251\!\cdots\!12p18T7+51932556467342037p27T8+132232612p36T9+p45T10 p^{18} T^{7} + 51932556467342037 p^{27} T^{8} + 132232612 p^{36} T^{9} + p^{45} T^{10}
67C2S5C_2 \wr S_5 1369388534T+167732159711342355T2 1 - 369388534 T + 167732159711342355 T^{2} - 38 ⁣ ⁣6438\!\cdots\!64T3+ T^{3} + 97 ⁣ ⁣5897\!\cdots\!58T4 T^{4} - 15 ⁣ ⁣4015\!\cdots\!40T5+ T^{5} + 97 ⁣ ⁣5897\!\cdots\!58p9T6 p^{9} T^{6} - 38 ⁣ ⁣6438\!\cdots\!64p18T7+167732159711342355p27T8369388534p36T9+p45T10 p^{18} T^{7} + 167732159711342355 p^{27} T^{8} - 369388534 p^{36} T^{9} + p^{45} T^{10}
71C2S5C_2 \wr S_5 1+212150457T+126078700918544326T2+ 1 + 212150457 T + 126078700918544326 T^{2} + 34 ⁣ ⁣3534\!\cdots\!35T3+ T^{3} + 83 ⁣ ⁣8983\!\cdots\!89T4+ T^{4} + 22 ⁣ ⁣0822\!\cdots\!08T5+ T^{5} + 83 ⁣ ⁣8983\!\cdots\!89p9T6+ p^{9} T^{6} + 34 ⁣ ⁣3534\!\cdots\!35p18T7+126078700918544326p27T8+212150457p36T9+p45T10 p^{18} T^{7} + 126078700918544326 p^{27} T^{8} + 212150457 p^{36} T^{9} + p^{45} T^{10}
73C2S5C_2 \wr S_5 1+252729806T+113432726759207253T2+ 1 + 252729806 T + 113432726759207253 T^{2} + 15 ⁣ ⁣4015\!\cdots\!40T3+ T^{3} + 10 ⁣ ⁣5810\!\cdots\!58T4+ T^{4} + 17 ⁣ ⁣2817\!\cdots\!28T5+ T^{5} + 10 ⁣ ⁣5810\!\cdots\!58p9T6+ p^{9} T^{6} + 15 ⁣ ⁣4015\!\cdots\!40p18T7+113432726759207253p27T8+252729806p36T9+p45T10 p^{18} T^{7} + 113432726759207253 p^{27} T^{8} + 252729806 p^{36} T^{9} + p^{45} T^{10}
79C2S5C_2 \wr S_5 11247271728T+1135234985129373579T2 1 - 1247271728 T + 1135234985129373579 T^{2} - 68 ⁣ ⁣9668\!\cdots\!96T3+ T^{3} + 33 ⁣ ⁣9433\!\cdots\!94T4 T^{4} - 12 ⁣ ⁣9212\!\cdots\!92T5+ T^{5} + 33 ⁣ ⁣9433\!\cdots\!94p9T6 p^{9} T^{6} - 68 ⁣ ⁣9668\!\cdots\!96p18T7+1135234985129373579p27T81247271728p36T9+p45T10 p^{18} T^{7} + 1135234985129373579 p^{27} T^{8} - 1247271728 p^{36} T^{9} + p^{45} T^{10}
83C2S5C_2 \wr S_5 1+1696894296T+1900005505469354847T2+ 1 + 1696894296 T + 1900005505469354847 T^{2} + 14 ⁣ ⁣2014\!\cdots\!20T3+ T^{3} + 90 ⁣ ⁣8290\!\cdots\!82T4+ T^{4} + 43 ⁣ ⁣1643\!\cdots\!16T5+ T^{5} + 90 ⁣ ⁣8290\!\cdots\!82p9T6+ p^{9} T^{6} + 14 ⁣ ⁣2014\!\cdots\!20p18T7+1900005505469354847p27T8+1696894296p36T9+p45T10 p^{18} T^{7} + 1900005505469354847 p^{27} T^{8} + 1696894296 p^{36} T^{9} + p^{45} T^{10}
89C2S5C_2 \wr S_5 1+753854382T+1391064645589335141T2+ 1 + 753854382 T + 1391064645589335141 T^{2} + 88 ⁣ ⁣0888\!\cdots\!08T3+ T^{3} + 90 ⁣ ⁣0690\!\cdots\!06T4+ T^{4} + 43 ⁣ ⁣0043\!\cdots\!00T5+ T^{5} + 90 ⁣ ⁣0690\!\cdots\!06p9T6+ p^{9} T^{6} + 88 ⁣ ⁣0888\!\cdots\!08p18T7+1391064645589335141p27T8+753854382p36T9+p45T10 p^{18} T^{7} + 1391064645589335141 p^{27} T^{8} + 753854382 p^{36} T^{9} + p^{45} T^{10}
97C2S5C_2 \wr S_5 13824606T+1359157730156522205T2+ 1 - 3824606 T + 1359157730156522205 T^{2} + 14 ⁣ ⁣2414\!\cdots\!24T3+ T^{3} + 16 ⁣ ⁣1416\!\cdots\!14T4+ T^{4} + 77 ⁣ ⁣5277\!\cdots\!52T5+ T^{5} + 16 ⁣ ⁣1416\!\cdots\!14p9T6+ p^{9} T^{6} + 14 ⁣ ⁣2414\!\cdots\!24p18T7+1359157730156522205p27T83824606p36T9+p45T10 p^{18} T^{7} + 1359157730156522205 p^{27} T^{8} - 3824606 p^{36} T^{9} + p^{45} T^{10}
show more
show less
   L(s)=p j=110(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.36462749398838407803608042422, −6.11263202233780018088132064070, −6.11085580338786371832332775452, −6.05247961861548853021674216887, −5.90425844863114287575812582517, −5.33480214382167014845621576701, −5.30524403864015557455964673795, −5.23371043423504310761354920702, −5.08359343174844389881588039583, −4.61478274580712744175322330520, −4.53054225710761360075774885240, −3.99912029286932311295399467493, −3.75829384685768785142009366180, −3.70088849799796462963140336257, −3.38678278814354224273486520182, −2.82386876762059450946972992543, −2.74414690373965449776635639408, −2.72932067970497987323170898050, −2.66579865352240529894891561831, −2.16573680279845600868060409414, −1.73102366334286269743387912110, −1.58865825813432691043483875373, −1.26658690714945075218574020483, −1.13933177173345191038160784256, −0.816540532126358517243907746039, 0, 0, 0, 0, 0, 0.816540532126358517243907746039, 1.13933177173345191038160784256, 1.26658690714945075218574020483, 1.58865825813432691043483875373, 1.73102366334286269743387912110, 2.16573680279845600868060409414, 2.66579865352240529894891561831, 2.72932067970497987323170898050, 2.74414690373965449776635639408, 2.82386876762059450946972992543, 3.38678278814354224273486520182, 3.70088849799796462963140336257, 3.75829384685768785142009366180, 3.99912029286932311295399467493, 4.53054225710761360075774885240, 4.61478274580712744175322330520, 5.08359343174844389881588039583, 5.23371043423504310761354920702, 5.30524403864015557455964673795, 5.33480214382167014845621576701, 5.90425844863114287575812582517, 6.05247961861548853021674216887, 6.11085580338786371832332775452, 6.11263202233780018088132064070, 6.36462749398838407803608042422

Graph of the ZZ-function along the critical line