↑

Properties

Label 6-5220e3-1.1-c1e3-0-1
Degree 66
Conductor 142236648000142236648000
Sign −1-1
Analytic cond. 72417.372417.3
Root an. cond. 6.456156.45615
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 33

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 4·7-s − 8·11-s − 2·13-s − 2·17-s + 8·19-s + 6·25-s + 3·29-s + 4·31-s − 12·35-s − 10·37-s − 10·41-s + 14·43-s + 10·47-s − 49-s − 10·53-s − 24·55-s − 8·59-s − 22·61-s − 6·65-s − 12·67-s + 8·71-s − 2·73-s + 32·77-s − 12·83-s − 6·85-s − 18·89-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.51·7-s − 2.41·11-s − 0.554·13-s − 0.485·17-s + 1.83·19-s + 6/5·25-s + 0.557·29-s + 0.718·31-s − 2.02·35-s − 1.64·37-s − 1.56·41-s + 2.13·43-s + 1.45·47-s − 1/7·49-s − 1.37·53-s − 3.23·55-s − 1.04·59-s − 2.81·61-s − 0.744·65-s − 1.46·67-s + 0.949·71-s − 0.234·73-s + 3.64·77-s − 1.31·83-s − 0.650·85-s − 1.90·89-s + ⋯

Functional equation

Λ(s)=((26⋅36⋅53⋅293)s/2 ΓC(s)3 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}
Λ(s)=((26⋅36⋅53⋅293)s/2 ΓC(s+1/2)3 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 26⋅36⋅53⋅2932^{6} \cdot 3^{6} \cdot 5^{3} \cdot 29^{3}
Sign: −1-1
Analytic conductor: 72417.372417.3
Root analytic conductor: 6.456156.45615
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 33
Selberg data: (6, 26⋅36⋅53⋅293, ( :1/2,1/2,1/2), −1)(6,\ 2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 29^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2 1 1
3 1 1
5C1C_1 (1−T)3 ( 1 - T )^{3}
29C1C_1 (1−T)3 ( 1 - T )^{3}
good7S4×C2S_4\times C_2 1+4T+17T2+52T3+17pT4+4p2T5+p3T6 1 + 4 T + 17 T^{2} + 52 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1+8T+45T2+164T3+45pT4+8p2T5+p3T6 1 + 8 T + 45 T^{2} + 164 T^{3} + 45 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1+2T+19T2+28T3+19pT4+2p2T5+p3T6 1 + 2 T + 19 T^{2} + 28 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1+2T+15T2−40T3+15pT4+2p2T5+p3T6 1 + 2 T + 15 T^{2} - 40 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
19S4×C2S_4\times C_2 1−8T+41T2−140T3+41pT4−8p2T5+p3T6 1 - 8 T + 41 T^{2} - 140 T^{3} + 41 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1+45T2+36T3+45pT4+p3T6 1 + 45 T^{2} + 36 T^{3} + 45 p T^{4} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−4T+61T2−284T3+61pT4−4p2T5+p3T6 1 - 4 T + 61 T^{2} - 284 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+10T+59T2+232T3+59pT4+10p2T5+p3T6 1 + 10 T + 59 T^{2} + 232 T^{3} + 59 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1+10T+135T2+796T3+135pT4+10p2T5+p3T6 1 + 10 T + 135 T^{2} + 796 T^{3} + 135 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}
43D6D_{6} 1−14T+113T2−656T3+113pT4−14p2T5+p3T6 1 - 14 T + 113 T^{2} - 656 T^{3} + 113 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1−10T+165T2−952T3+165pT4−10p2T5+p3T6 1 - 10 T + 165 T^{2} - 952 T^{3} + 165 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+10T+147T2+988T3+147pT4+10p2T5+p3T6 1 + 10 T + 147 T^{2} + 988 T^{3} + 147 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1+8T+9T2−544T3+9pT4+8p2T5+p3T6 1 + 8 T + 9 T^{2} - 544 T^{3} + 9 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
61D6D_{6} 1+22T+299T2+2788T3+299pT4+22p2T5+p3T6 1 + 22 T + 299 T^{2} + 2788 T^{3} + 299 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1+12T+225T2+1540T3+225pT4+12p2T5+p3T6 1 + 12 T + 225 T^{2} + 1540 T^{3} + 225 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1−8T+189T2−992T3+189pT4−8p2T5+p3T6 1 - 8 T + 189 T^{2} - 992 T^{3} + 189 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1+2T+139T2−32T3+139pT4+2p2T5+p3T6 1 + 2 T + 139 T^{2} - 32 T^{3} + 139 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1+129T2+244T3+129pT4+p3T6 1 + 129 T^{2} + 244 T^{3} + 129 p T^{4} + p^{3} T^{6}
83S4×C2S_4\times C_2 1+12T+213T2+1884T3+213pT4+12p2T5+p3T6 1 + 12 T + 213 T^{2} + 1884 T^{3} + 213 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+18T+279T2+3132T3+279pT4+18p2T5+p3T6 1 + 18 T + 279 T^{2} + 3132 T^{3} + 279 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+38T+763T2+9280T3+763pT4+38p2T5+p3T6 1 + 38 T + 763 T^{2} + 9280 T^{3} + 763 p T^{4} + 38 p^{2} T^{5} + p^{3} T^{6}
show more
show less
   L(s)=∏p ∏j=16(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.59856827358761333804860977260, −7.35118778127331763399440308823, −7.14200907682236930242847764227, −6.92687192791166052311481522380, −6.56033229582637809809321863363, −6.34785680134332272186749732621, −6.26439082449963576541415048603, −5.95375970594210402773554882291, −5.51592119291581419662411843173, −5.42716492047027328077659318231, −5.26283226594296558626517902941, −5.01405023583652197521142122963, −4.92846684637236090465011592352, −4.34484038441904270340928245055, −4.07560427281879546221822867948, −3.99316201191640441343857693944, −3.31387031175211363420609494670, −3.05966648970538958709406427185, −2.95193618032618234049668460411, −2.68416228837162531807896907641, −2.54456662855585889951573625539, −2.34865965233774095571105588383, −1.44087116678613878655112741439, −1.42096228406856806185233043419, −1.27692092150067006038620236935, 0, 0, 0, 1.27692092150067006038620236935, 1.42096228406856806185233043419, 1.44087116678613878655112741439, 2.34865965233774095571105588383, 2.54456662855585889951573625539, 2.68416228837162531807896907641, 2.95193618032618234049668460411, 3.05966648970538958709406427185, 3.31387031175211363420609494670, 3.99316201191640441343857693944, 4.07560427281879546221822867948, 4.34484038441904270340928245055, 4.92846684637236090465011592352, 5.01405023583652197521142122963, 5.26283226594296558626517902941, 5.42716492047027328077659318231, 5.51592119291581419662411843173, 5.95375970594210402773554882291, 6.26439082449963576541415048603, 6.34785680134332272186749732621, 6.56033229582637809809321863363, 6.92687192791166052311481522380, 7.14200907682236930242847764227, 7.35118778127331763399440308823, 7.59856827358761333804860977260

Graph of the ZZ-function along the critical line