Properties

Label 8-45e4-1.1-c21e4-0-0
Degree 88
Conductor 41006254100625
Sign 11
Analytic cond. 2.50170×1082.50170\times 10^{8}
Root an. cond. 11.214411.2144
Motivic weight 2121
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  + 897·2-s − 4.37e6·4-s − 3.90e7·5-s − 2.34e8·7-s − 4.81e9·8-s − 3.50e10·10-s − 3.14e10·11-s − 2.70e10·13-s − 2.10e11·14-s + 6.70e12·16-s + 2.94e12·17-s − 2.42e13·19-s + 1.70e14·20-s − 2.82e13·22-s − 1.03e13·23-s + 9.53e14·25-s − 2.42e13·26-s + 1.02e15·28-s − 4.72e15·29-s − 1.09e15·31-s + 8.96e15·32-s + 2.64e15·34-s + 9.16e15·35-s + 4.81e15·37-s − 2.17e16·38-s + 1.88e17·40-s − 2.89e17·41-s + ⋯
L(s)  = 1  + 0.619·2-s − 2.08·4-s − 1.78·5-s − 0.313·7-s − 1.58·8-s − 1.10·10-s − 0.366·11-s − 0.0543·13-s − 0.194·14-s + 1.52·16-s + 0.354·17-s − 0.908·19-s + 3.73·20-s − 0.226·22-s − 0.0520·23-s + 2·25-s − 0.0336·26-s + 0.654·28-s − 2.08·29-s − 0.239·31-s + 1.40·32-s + 0.219·34-s + 0.561·35-s + 0.164·37-s − 0.562·38-s + 2.83·40-s − 3.37·41-s + ⋯

Functional equation

Λ(s)=(4100625s/2ΓC(s)4L(s)=(Λ(22s)\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}
Λ(s)=(4100625s/2ΓC(s+21/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+21/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 41006254100625    =    38543^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 2.50170×1082.50170\times 10^{8}
Root analytic conductor: 11.214411.2144
Motivic weight: 2121
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 4100625, ( :21/2,21/2,21/2,21/2), 1)(8,\ 4100625,\ (\ :21/2, 21/2, 21/2, 21/2),\ 1)

Particular Values

L(11)L(11) \approx 0.019208766670.01920876667
L(12)L(\frac12) \approx 0.019208766670.01920876667
L(232)L(\frac{23}{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)
bad3 1 1
5C1C_1 (1+p10T)4 ( 1 + p^{10} T )^{4}
good2C2S4C_2 \wr S_4 1897T+2589085pT2117236669p5T3+14637391839p10T4117236669p26T5+2589085p43T6897p63T7+p84T8 1 - 897 T + 2589085 p T^{2} - 117236669 p^{5} T^{3} + 14637391839 p^{10} T^{4} - 117236669 p^{26} T^{5} + 2589085 p^{43} T^{6} - 897 p^{63} T^{7} + p^{84} T^{8}
7C2S4C_2 \wr S_4 1+4787296p2T+7592135803176028p2T2 1 + 4787296 p^{2} T + 7592135803176028 p^{2} T^{2} - 13 ⁣ ⁣3613\!\cdots\!36p4T3+ p^{4} T^{3} + 73 ⁣ ⁣1073\!\cdots\!10p4T4 p^{4} T^{4} - 13 ⁣ ⁣3613\!\cdots\!36p25T5+7592135803176028p44T6+4787296p65T7+p84T8 p^{25} T^{5} + 7592135803176028 p^{44} T^{6} + 4787296 p^{65} T^{7} + p^{84} T^{8}
11C2S4C_2 \wr S_4 1+31491830256T+ 1 + 31491830256 T + 93 ⁣ ⁣9293\!\cdots\!92pT2+ p T^{2} + 46 ⁣ ⁣0046\!\cdots\!00p2T3+ p^{2} T^{3} + 53 ⁣ ⁣0653\!\cdots\!06p3T4+ p^{3} T^{4} + 46 ⁣ ⁣0046\!\cdots\!00p23T5+ p^{23} T^{5} + 93 ⁣ ⁣9293\!\cdots\!92p43T6+31491830256p63T7+p84T8 p^{43} T^{6} + 31491830256 p^{63} T^{7} + p^{84} T^{8}
13C2S4C_2 \wr S_4 1+27017977768T 1 + 27017977768 T - 92 ⁣ ⁣7292\!\cdots\!72pT2+ p T^{2} + 33 ⁣ ⁣2433\!\cdots\!24p2T3+ p^{2} T^{3} + 39 ⁣ ⁣3039\!\cdots\!30p3T4+ p^{3} T^{4} + 33 ⁣ ⁣2433\!\cdots\!24p23T5 p^{23} T^{5} - 92 ⁣ ⁣7292\!\cdots\!72p43T6+27017977768p63T7+p84T8 p^{43} T^{6} + 27017977768 p^{63} T^{7} + p^{84} T^{8}
17C2S4C_2 \wr S_4 12946095028888T+ 1 - 2946095028888 T + 67 ⁣ ⁣1667\!\cdots\!16pT2 p T^{2} - 25 ⁣ ⁣2025\!\cdots\!20p2T3+ p^{2} T^{3} + 19 ⁣ ⁣0219\!\cdots\!02p3T4 p^{3} T^{4} - 25 ⁣ ⁣2025\!\cdots\!20p23T5+ p^{23} T^{5} + 67 ⁣ ⁣1667\!\cdots\!16p43T62946095028888p63T7+p84T8 p^{43} T^{6} - 2946095028888 p^{63} T^{7} + p^{84} T^{8}
19C2S4C_2 \wr S_4 1+24270353300752T+ 1 + 24270353300752 T + 48 ⁣ ⁣9248\!\cdots\!92p2T2+ p^{2} T^{2} + 36 ⁣ ⁣3636\!\cdots\!36p3T3+ p^{3} T^{3} + 21 ⁣ ⁣2621\!\cdots\!26p3T4+ p^{3} T^{4} + 36 ⁣ ⁣3636\!\cdots\!36p24T5+ p^{24} T^{5} + 48 ⁣ ⁣9248\!\cdots\!92p44T6+24270353300752p63T7+p84T8 p^{44} T^{6} + 24270353300752 p^{63} T^{7} + p^{84} T^{8}
23C2S4C_2 \wr S_4 1+10350924920928T+ 1 + 10350924920928 T + 70 ⁣ ⁣0870\!\cdots\!08T2+ T^{2} + 70 ⁣ ⁣1270\!\cdots\!12T3+ T^{3} + 23 ⁣ ⁣1023\!\cdots\!10T4+ T^{4} + 70 ⁣ ⁣1270\!\cdots\!12p21T5+ p^{21} T^{5} + 70 ⁣ ⁣0870\!\cdots\!08p42T6+10350924920928p63T7+p84T8 p^{42} T^{6} + 10350924920928 p^{63} T^{7} + p^{84} T^{8}
29C2S4C_2 \wr S_4 1+4728924677079096T+ 1 + 4728924677079096 T + 11 ⁣ ⁣0011\!\cdots\!00T2+ T^{2} + 17 ⁣ ⁣9217\!\cdots\!92T3+ T^{3} + 17 ⁣ ⁣1817\!\cdots\!18T4+ T^{4} + 17 ⁣ ⁣9217\!\cdots\!92p21T5+ p^{21} T^{5} + 11 ⁣ ⁣0011\!\cdots\!00p42T6+4728924677079096p63T7+p84T8 p^{42} T^{6} + 4728924677079096 p^{63} T^{7} + p^{84} T^{8}
31C2S4C_2 \wr S_4 1+1094923910405536T+ 1 + 1094923910405536 T + 68 ⁣ ⁣4868\!\cdots\!48T2+ T^{2} + 44 ⁣ ⁣4844\!\cdots\!48T3+ T^{3} + 19 ⁣ ⁣5419\!\cdots\!54T4+ T^{4} + 44 ⁣ ⁣4844\!\cdots\!48p21T5+ p^{21} T^{5} + 68 ⁣ ⁣4868\!\cdots\!48p42T6+1094923910405536p63T7+p84T8 p^{42} T^{6} + 1094923910405536 p^{63} T^{7} + p^{84} T^{8}
37C2S4C_2 \wr S_4 14813435696247096T+ 1 - 4813435696247096 T + 17 ⁣ ⁣1217\!\cdots\!12T2 T^{2} - 33 ⁣ ⁣3633\!\cdots\!36T3+ T^{3} + 15 ⁣ ⁣5015\!\cdots\!50T4 T^{4} - 33 ⁣ ⁣3633\!\cdots\!36p21T5+ p^{21} T^{5} + 17 ⁣ ⁣1217\!\cdots\!12p42T64813435696247096p63T7+p84T8 p^{42} T^{6} - 4813435696247096 p^{63} T^{7} + p^{84} T^{8}
41C2S4C_2 \wr S_4 1+289731591445930344T+ 1 + 289731591445930344 T + 49 ⁣ ⁣6849\!\cdots\!68T2+ T^{2} + 57 ⁣ ⁣5257\!\cdots\!52T3+ T^{3} + 55 ⁣ ⁣1455\!\cdots\!14T4+ T^{4} + 57 ⁣ ⁣5257\!\cdots\!52p21T5+ p^{21} T^{5} + 49 ⁣ ⁣6849\!\cdots\!68p42T6+289731591445930344p63T7+p84T8 p^{42} T^{6} + 289731591445930344 p^{63} T^{7} + p^{84} T^{8}
43C2S4C_2 \wr S_4 1451091912658458000T+ 1 - 451091912658458000 T + 13 ⁣ ⁣0013\!\cdots\!00T2 T^{2} - 27 ⁣ ⁣0027\!\cdots\!00T3+ T^{3} + 44 ⁣ ⁣9844\!\cdots\!98T4 T^{4} - 27 ⁣ ⁣0027\!\cdots\!00p21T5+ p^{21} T^{5} + 13 ⁣ ⁣0013\!\cdots\!00p42T6451091912658458000p63T7+p84T8 p^{42} T^{6} - 451091912658458000 p^{63} T^{7} + p^{84} T^{8}
47C2S4C_2 \wr S_4 1+813883435638492480T+ 1 + 813883435638492480 T + 59 ⁣ ⁣2059\!\cdots\!20T2+ T^{2} + 27 ⁣ ⁣6027\!\cdots\!60T3+ T^{3} + 11 ⁣ ⁣1811\!\cdots\!18T4+ T^{4} + 27 ⁣ ⁣6027\!\cdots\!60p21T5+ p^{21} T^{5} + 59 ⁣ ⁣2059\!\cdots\!20p42T6+813883435638492480p63T7+p84T8 p^{42} T^{6} + 813883435638492480 p^{63} T^{7} + p^{84} T^{8}
53C2S4C_2 \wr S_4 1+697278335404085208T+ 1 + 697278335404085208 T + 47 ⁣ ⁣2847\!\cdots\!28T2+ T^{2} + 22 ⁣ ⁣7222\!\cdots\!72T3+ T^{3} + 10 ⁣ ⁣5010\!\cdots\!50T4+ T^{4} + 22 ⁣ ⁣7222\!\cdots\!72p21T5+ p^{21} T^{5} + 47 ⁣ ⁣2847\!\cdots\!28p42T6+697278335404085208p63T7+p84T8 p^{42} T^{6} + 697278335404085208 p^{63} T^{7} + p^{84} T^{8}
59C2S4C_2 \wr S_4 1+6622888614569598192T+ 1 + 6622888614569598192 T + 40 ⁣ ⁣7240\!\cdots\!72T2+ T^{2} + 20 ⁣ ⁣0420\!\cdots\!04T3+ T^{3} + 88 ⁣ ⁣3488\!\cdots\!34T4+ T^{4} + 20 ⁣ ⁣0420\!\cdots\!04p21T5+ p^{21} T^{5} + 40 ⁣ ⁣7240\!\cdots\!72p42T6+6622888614569598192p63T7+p84T8 p^{42} T^{6} + 6622888614569598192 p^{63} T^{7} + p^{84} T^{8}
61C2S4C_2 \wr S_4 17390887218011683320T+ 1 - 7390887218011683320 T + 30 ⁣ ⁣9630\!\cdots\!96T2 T^{2} - 27 ⁣ ⁣8027\!\cdots\!80T3+ T^{3} + 27 ⁣ ⁣4627\!\cdots\!46T4 T^{4} - 27 ⁣ ⁣8027\!\cdots\!80p21T5+ p^{21} T^{5} + 30 ⁣ ⁣9630\!\cdots\!96p42T67390887218011683320p63T7+p84T8 p^{42} T^{6} - 7390887218011683320 p^{63} T^{7} + p^{84} T^{8}
67C2S4C_2 \wr S_4 1+24188188449376788688T+ 1 + 24188188449376788688 T + 66 ⁣ ⁣7266\!\cdots\!72T2+ T^{2} + 12 ⁣ ⁣8012\!\cdots\!80T3+ T^{3} + 22 ⁣ ⁣2622\!\cdots\!26T4+ T^{4} + 12 ⁣ ⁣8012\!\cdots\!80p21T5+ p^{21} T^{5} + 66 ⁣ ⁣7266\!\cdots\!72p42T6+24188188449376788688p63T7+p84T8 p^{42} T^{6} + 24188188449376788688 p^{63} T^{7} + p^{84} T^{8}
71C2S4C_2 \wr S_4 137390337803999713312T+ 1 - 37390337803999713312 T + 18 ⁣ ⁣8818\!\cdots\!88T2 T^{2} - 49 ⁣ ⁣6449\!\cdots\!64T3+ T^{3} + 15 ⁣ ⁣7015\!\cdots\!70T4 T^{4} - 49 ⁣ ⁣6449\!\cdots\!64p21T5+ p^{21} T^{5} + 18 ⁣ ⁣8818\!\cdots\!88p42T637390337803999713312p63T7+p84T8 p^{42} T^{6} - 37390337803999713312 p^{63} T^{7} + p^{84} T^{8}
73C2S4C_2 \wr S_4 1+37253672904265201432T+ 1 + 37253672904265201432 T + 53 ⁣ ⁣2853\!\cdots\!28T2+ T^{2} + 13 ⁣ ⁣8813\!\cdots\!88T3+ T^{3} + 10 ⁣ ⁣3010\!\cdots\!30T4+ T^{4} + 13 ⁣ ⁣8813\!\cdots\!88p21T5+ p^{21} T^{5} + 53 ⁣ ⁣2853\!\cdots\!28p42T6+37253672904265201432p63T7+p84T8 p^{42} T^{6} + 37253672904265201432 p^{63} T^{7} + p^{84} T^{8}
79C2S4C_2 \wr S_4 1+ 1 + 10 ⁣ ⁣0010\!\cdots\!00T+ T + 20 ⁣ ⁣1620\!\cdots\!16T2+ T^{2} + 13 ⁣ ⁣0013\!\cdots\!00T3+ T^{3} + 17 ⁣ ⁣4617\!\cdots\!46T4+ T^{4} + 13 ⁣ ⁣0013\!\cdots\!00p21T5+ p^{21} T^{5} + 20 ⁣ ⁣1620\!\cdots\!16p42T6+ p^{42} T^{6} + 10 ⁣ ⁣0010\!\cdots\!00p63T7+p84T8 p^{63} T^{7} + p^{84} T^{8}
83C2S4C_2 \wr S_4 1 1 - 14 ⁣ ⁣9614\!\cdots\!96T+ T + 49 ⁣ ⁣6049\!\cdots\!60T2 T^{2} - 54 ⁣ ⁣5654\!\cdots\!56T3+ T^{3} + 11 ⁣ ⁣2611\!\cdots\!26T4 T^{4} - 54 ⁣ ⁣5654\!\cdots\!56p21T5+ p^{21} T^{5} + 49 ⁣ ⁣6049\!\cdots\!60p42T6 p^{42} T^{6} - 14 ⁣ ⁣9614\!\cdots\!96p63T7+p84T8 p^{63} T^{7} + p^{84} T^{8}
89C2S4C_2 \wr S_4 1+ 1 + 55 ⁣ ⁣4855\!\cdots\!48T+ T + 43 ⁣ ⁣5243\!\cdots\!52T2+ T^{2} + 14 ⁣ ⁣9614\!\cdots\!96T3+ T^{3} + 60 ⁣ ⁣1460\!\cdots\!14T4+ T^{4} + 14 ⁣ ⁣9614\!\cdots\!96p21T5+ p^{21} T^{5} + 43 ⁣ ⁣5243\!\cdots\!52p42T6+ p^{42} T^{6} + 55 ⁣ ⁣4855\!\cdots\!48p63T7+p84T8 p^{63} T^{7} + p^{84} T^{8}
97C2S4C_2 \wr S_4 1+ 1 + 23 ⁣ ⁣2823\!\cdots\!28T+ T + 13 ⁣ ⁣3213\!\cdots\!32T2+ T^{2} + 61 ⁣ ⁣2061\!\cdots\!20T3+ T^{3} + 80 ⁣ ⁣6680\!\cdots\!66T4+ T^{4} + 61 ⁣ ⁣2061\!\cdots\!20p21T5+ p^{21} T^{5} + 13 ⁣ ⁣3213\!\cdots\!32p42T6+ p^{42} T^{6} + 23 ⁣ ⁣2823\!\cdots\!28p63T7+p84T8 p^{63} T^{7} + p^{84} T^{8}
show more
show less
   L(s)=p j=18(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.925661808048971603484989154936, −7.47400900377655106709746858276, −7.11931413950061897356061873488, −6.71548747807764158673202294660, −6.69661646011760205096314348576, −6.08379063346739706529498170478, −5.49282207059025500609540872302, −5.48762596346637760272339649288, −5.37802426682904202908343187039, −4.64948348735575554198424198864, −4.55586248848710531257708936492, −4.28034506390677916071463978657, −4.25595363719694055877070508889, −3.83725941632738691745049331827, −3.34414659808914850725861468710, −3.21115415125424645186078196694, −3.20942321412360675521358032589, −2.63281472626392836416860681181, −1.98420057947417468079924546350, −1.67414911027982636189640623942, −1.63762515646515751208863686285, −0.838567829295619169433666960052, −0.50659759460686242462831000900, −0.44074888670186003444212388732, −0.03115564611230382885270971954, 0.03115564611230382885270971954, 0.44074888670186003444212388732, 0.50659759460686242462831000900, 0.838567829295619169433666960052, 1.63762515646515751208863686285, 1.67414911027982636189640623942, 1.98420057947417468079924546350, 2.63281472626392836416860681181, 3.20942321412360675521358032589, 3.21115415125424645186078196694, 3.34414659808914850725861468710, 3.83725941632738691745049331827, 4.25595363719694055877070508889, 4.28034506390677916071463978657, 4.55586248848710531257708936492, 4.64948348735575554198424198864, 5.37802426682904202908343187039, 5.48762596346637760272339649288, 5.49282207059025500609540872302, 6.08379063346739706529498170478, 6.69661646011760205096314348576, 6.71548747807764158673202294660, 7.11931413950061897356061873488, 7.47400900377655106709746858276, 7.925661808048971603484989154936

Graph of the ZZ-function along the critical line