↑

Properties

Label 8-1805e4-1.1-c1e4-0-1
Degree 88
Conductor 1.061×10131.061\times 10^{13}
Sign 11
Analytic cond. 43153.643153.6
Root an. cond. 3.796443.79644
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  + 2·2-s − 2·3-s + 2·4-s − 4·5-s − 4·6-s + 4·7-s + 4·8-s − 8·10-s + 4·11-s − 4·12-s − 2·13-s + 8·14-s + 8·15-s + 3·16-s + 4·17-s − 8·20-s − 8·21-s + 8·22-s − 8·23-s − 8·24-s + 10·25-s − 4·26-s + 6·27-s + 8·28-s − 4·29-s + 16·30-s − 4·31-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s − 1.63·6-s + 1.51·7-s + 1.41·8-s − 2.52·10-s + 1.20·11-s − 1.15·12-s − 0.554·13-s + 2.13·14-s + 2.06·15-s + 3/4·16-s + 0.970·17-s − 1.78·20-s − 1.74·21-s + 1.70·22-s − 1.66·23-s − 1.63·24-s + 2·25-s − 0.784·26-s + 1.15·27-s + 1.51·28-s − 0.742·29-s + 2.92·30-s − 0.718·31-s + ⋯

Functional equation

Λ(s)=((54⋅198)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((54⋅198)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 54⋅1985^{4} \cdot 19^{8}
Sign: 11
Analytic conductor: 43153.643153.6
Root analytic conductor: 3.796443.79644
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 54⋅198, ( :1/2,1/2,1/2,1/2), 1)(8,\ 5^{4} \cdot 19^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 4.4210610304.421061030
L(12)L(\frac12) ≈\approx 4.4210610304.421061030
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad5C1C_1 (1+T)4 ( 1 + T )^{4}
19 1 1
good2C2≀S4C_2 \wr S_4 1−pT+pT2−p2T3+9T4−p3T5+p3T6−p4T7+p4T8 1 - p T + p T^{2} - p^{2} T^{3} + 9 T^{4} - p^{3} T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8}
3C2≀S4C_2 \wr S_4 1+2T+4T2+2T3+2T4+2pT5+4p2T6+2p3T7+p4T8 1 + 2 T + 4 T^{2} + 2 T^{3} + 2 T^{4} + 2 p T^{5} + 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
7S4×C2S_4\times C_2 1−4T+12T2−36T3+102T4−36pT5+12p2T6−4p3T7+p4T8 1 - 4 T + 12 T^{2} - 36 T^{3} + 102 T^{4} - 36 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
11GL(2,3)\textrm{GL(2,3)} 1−4T+28T2−100T3+422T4−100pT5+28p2T6−4p3T7+p4T8 1 - 4 T + 28 T^{2} - 100 T^{3} + 422 T^{4} - 100 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
13C2≀S4C_2 \wr S_4 1+2T+28T2+46T3+410T4+46pT5+28p2T6+2p3T7+p4T8 1 + 2 T + 28 T^{2} + 46 T^{3} + 410 T^{4} + 46 p T^{5} + 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
17C2≀S4C_2 \wr S_4 1−4T+36T2−188T3+694T4−188pT5+36p2T6−4p3T7+p4T8 1 - 4 T + 36 T^{2} - 188 T^{3} + 694 T^{4} - 188 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
23C2≀S4C_2 \wr S_4 1+8T+68T2+376T3+2358T4+376pT5+68p2T6+8p3T7+p4T8 1 + 8 T + 68 T^{2} + 376 T^{3} + 2358 T^{4} + 376 p T^{5} + 68 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
29GL(2,3)\textrm{GL(2,3)} 1+4T+84T2+332T3+3238T4+332pT5+84p2T6+4p3T7+p4T8 1 + 4 T + 84 T^{2} + 332 T^{3} + 3238 T^{4} + 332 p T^{5} + 84 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
31C2≀S4C_2 \wr S_4 1+4T+44T2−140T3+166T4−140pT5+44p2T6+4p3T7+p4T8 1 + 4 T + 44 T^{2} - 140 T^{3} + 166 T^{4} - 140 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
37C2≀S4C_2 \wr S_4 1−6T+124T2−626T3+6442T4−626pT5+124p2T6−6p3T7+p4T8 1 - 6 T + 124 T^{2} - 626 T^{3} + 6442 T^{4} - 626 p T^{5} + 124 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}
41C2≀S4C_2 \wr S_4 1+16T+220T2+1936T3+14438T4+1936pT5+220p2T6+16p3T7+p4T8 1 + 16 T + 220 T^{2} + 1936 T^{3} + 14438 T^{4} + 1936 p T^{5} + 220 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8}
43C2≀S4C_2 \wr S_4 1−4T+156T2−468T3+9750T4−468pT5+156p2T6−4p3T7+p4T8 1 - 4 T + 156 T^{2} - 468 T^{3} + 9750 T^{4} - 468 p T^{5} + 156 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
47C2≀S4C_2 \wr S_4 1+12T+124T2+1036T3+8294T4+1036pT5+124p2T6+12p3T7+p4T8 1 + 12 T + 124 T^{2} + 1036 T^{3} + 8294 T^{4} + 1036 p T^{5} + 124 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
53C2≀S4C_2 \wr S_4 1−10T+4pT2−1406T3+16506T4−1406pT5+4p3T6−10p3T7+p4T8 1 - 10 T + 4 p T^{2} - 1406 T^{3} + 16506 T^{4} - 1406 p T^{5} + 4 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}
59C2≀S4C_2 \wr S_4 1+172T2+224T3+13142T4+224pT5+172p2T6+p4T8 1 + 172 T^{2} + 224 T^{3} + 13142 T^{4} + 224 p T^{5} + 172 p^{2} T^{6} + p^{4} T^{8}
61C2≀S4C_2 \wr S_4 1−20T+300T2−2972T3+26502T4−2972pT5+300p2T6−20p3T7+p4T8 1 - 20 T + 300 T^{2} - 2972 T^{3} + 26502 T^{4} - 2972 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}
67C2≀S4C_2 \wr S_4 1−18T+276T2−3130T3+26930T4−3130pT5+276p2T6−18p3T7+p4T8 1 - 18 T + 276 T^{2} - 3130 T^{3} + 26930 T^{4} - 3130 p T^{5} + 276 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8}
71C2≀S4C_2 \wr S_4 1−20T+316T2−3236T3+30566T4−3236pT5+316p2T6−20p3T7+p4T8 1 - 20 T + 316 T^{2} - 3236 T^{3} + 30566 T^{4} - 3236 p T^{5} + 316 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}
73C2≀S4C_2 \wr S_4 1−28T+548T2−6916T3+69526T4−6916pT5+548p2T6−28p3T7+p4T8 1 - 28 T + 548 T^{2} - 6916 T^{3} + 69526 T^{4} - 6916 p T^{5} + 548 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8}
79C2≀S4C_2 \wr S_4 1−16T+348T2−3312T3+40646T4−3312pT5+348p2T6−16p3T7+p4T8 1 - 16 T + 348 T^{2} - 3312 T^{3} + 40646 T^{4} - 3312 p T^{5} + 348 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8}
83C2≀S4C_2 \wr S_4 1+260T2−112T3+29862T4−112pT5+260p2T6+p4T8 1 + 260 T^{2} - 112 T^{3} + 29862 T^{4} - 112 p T^{5} + 260 p^{2} T^{6} + p^{4} T^{8}
89C2≀S4C_2 \wr S_4 1+4T+212T2+1244T3+22134T4+1244pT5+212p2T6+4p3T7+p4T8 1 + 4 T + 212 T^{2} + 1244 T^{3} + 22134 T^{4} + 1244 p T^{5} + 212 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
97C2≀S4C_2 \wr S_4 1+30T+612T2+8738T3+98522T4+8738pT5+612p2T6+30p3T7+p4T8 1 + 30 T + 612 T^{2} + 8738 T^{3} + 98522 T^{4} + 8738 p T^{5} + 612 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8}
show more
show less
   L(s)=∏p ∏j=18(1−αj,p p−s)−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

−6.61054074024909909076916988385, −6.55467413009441385029304705225, −5.99335280573231450040210584931, −5.86350402135829098367204157809, −5.65551111786547195104466626618, −5.39115809748685932343941940298, −5.09483183582300447916141946734, −5.07002820458782252301618911480, −5.00474620690255082167397730928, −4.73561164895114785653516741704, −4.45874657319188368051399772900, −4.11937921977363612826867595336, −3.93777180878089654692540093788, −3.89502231656176158638969992703, −3.65530108400113259204446361768, −3.50686542365179419950908498446, −3.21593653361583710890314896059, −2.57734648814662431764233080693, −2.57607082068416537821561297740, −1.97221363531484072848400416088, −1.79377811474673260537343947752, −1.78127667405207560327355660647, −1.04579847525429894475470299308, −0.71535563597378505722975222113, −0.42114664627238455157593270570, 0.42114664627238455157593270570, 0.71535563597378505722975222113, 1.04579847525429894475470299308, 1.78127667405207560327355660647, 1.79377811474673260537343947752, 1.97221363531484072848400416088, 2.57607082068416537821561297740, 2.57734648814662431764233080693, 3.21593653361583710890314896059, 3.50686542365179419950908498446, 3.65530108400113259204446361768, 3.89502231656176158638969992703, 3.93777180878089654692540093788, 4.11937921977363612826867595336, 4.45874657319188368051399772900, 4.73561164895114785653516741704, 5.00474620690255082167397730928, 5.07002820458782252301618911480, 5.09483183582300447916141946734, 5.39115809748685932343941940298, 5.65551111786547195104466626618, 5.86350402135829098367204157809, 5.99335280573231450040210584931, 6.55467413009441385029304705225, 6.61054074024909909076916988385

Graph of the ZZ-function along the critical line