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Properties

Label 6-1472e3-1.1-c3e3-0-2
Degree 66
Conductor 31895060483189506048
Sign 11
Analytic cond. 655121.655121.
Root an. cond. 9.319379.31937
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 16·5-s + 18·7-s − 31·9-s − 40·11-s + 95·13-s + 16·15-s − 42·17-s − 26·19-s − 18·21-s − 69·23-s + 157·25-s − 14·27-s + 181·29-s + 705·31-s + 40·33-s − 288·35-s + 80·37-s − 95·39-s + 281·41-s − 248·43-s + 496·45-s + 517·47-s − 369·49-s + 42·51-s + 190·53-s + 640·55-s + ⋯
L(s)  = 1  − 0.192·3-s − 1.43·5-s + 0.971·7-s − 1.14·9-s − 1.09·11-s + 2.02·13-s + 0.275·15-s − 0.599·17-s − 0.313·19-s − 0.187·21-s − 0.625·23-s + 1.25·25-s − 0.0997·27-s + 1.15·29-s + 4.08·31-s + 0.211·33-s − 1.39·35-s + 0.355·37-s − 0.390·39-s + 1.07·41-s − 0.879·43-s + 1.64·45-s + 1.60·47-s − 1.07·49-s + 0.115·51-s + 0.492·53-s + 1.56·55-s + ⋯

Functional equation

Λ(s)=((218⋅233)s/2 ΓC(s)3 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((218⋅233)s/2 ΓC(s+3/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 218⋅2332^{18} \cdot 23^{3}
Sign: 11
Analytic conductor: 655121.655121.
Root analytic conductor: 9.319379.31937
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 218⋅233, ( :3/2,3/2,3/2), 1)(6,\ 2^{18} \cdot 23^{3} ,\ ( \ : 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 1.1046286101.104628610
L(12)L(\frac12) ≈\approx 1.1046286101.104628610
L(52)L(\frac{5}{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
23C1C_1 (1+pT)3 ( 1 + p T )^{3}
good3S4×C2S_4\times C_2 1+T+32T2+77T3+32p3T4+p6T5+p9T6 1 + T + 32 T^{2} + 77 T^{3} + 32 p^{3} T^{4} + p^{6} T^{5} + p^{9} T^{6}
5S4×C2S_4\times C_2 1+16T+99T2−232T3+99p3T4+16p6T5+p9T6 1 + 16 T + 99 T^{2} - 232 T^{3} + 99 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6}
7S4×C2S_4\times C_2 1−18T+99pT2−10964T3+99p4T4−18p6T5+p9T6 1 - 18 T + 99 p T^{2} - 10964 T^{3} + 99 p^{4} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6}
11S4×C2S_4\times C_2 1+40T+1725T2+40424T3+1725p3T4+40p6T5+p9T6 1 + 40 T + 1725 T^{2} + 40424 T^{3} + 1725 p^{3} T^{4} + 40 p^{6} T^{5} + p^{9} T^{6}
13S4×C2S_4\times C_2 1−95T+7978T2−385251T3+7978p3T4−95p6T5+p9T6 1 - 95 T + 7978 T^{2} - 385251 T^{3} + 7978 p^{3} T^{4} - 95 p^{6} T^{5} + p^{9} T^{6}
17S4×C2S_4\times C_2 1+42T+12523T2+356236T3+12523p3T4+42p6T5+p9T6 1 + 42 T + 12523 T^{2} + 356236 T^{3} + 12523 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6}
19S4×C2S_4\times C_2 1+26T+1325T2−787148T3+1325p3T4+26p6T5+p9T6 1 + 26 T + 1325 T^{2} - 787148 T^{3} + 1325 p^{3} T^{4} + 26 p^{6} T^{5} + p^{9} T^{6}
29S4×C2S_4\times C_2 1−181T+32658T2−1717337T3+32658p3T4−181p6T5+p9T6 1 - 181 T + 32658 T^{2} - 1717337 T^{3} + 32658 p^{3} T^{4} - 181 p^{6} T^{5} + p^{9} T^{6}
31S4×C2S_4\times C_2 1−705T+244660T2−52242909T3+244660p3T4−705p6T5+p9T6 1 - 705 T + 244660 T^{2} - 52242909 T^{3} + 244660 p^{3} T^{4} - 705 p^{6} T^{5} + p^{9} T^{6}
37S4×C2S_4\times C_2 1−80T+28931T2+11253064T3+28931p3T4−80p6T5+p9T6 1 - 80 T + 28931 T^{2} + 11253064 T^{3} + 28931 p^{3} T^{4} - 80 p^{6} T^{5} + p^{9} T^{6}
41S4×C2S_4\times C_2 1−281T+3494pT2−41307101T3+3494p4T4−281p6T5+p9T6 1 - 281 T + 3494 p T^{2} - 41307101 T^{3} + 3494 p^{4} T^{4} - 281 p^{6} T^{5} + p^{9} T^{6}
43S4×C2S_4\times C_2 1+248T+152377T2+36661968T3+152377p3T4+248p6T5+p9T6 1 + 248 T + 152377 T^{2} + 36661968 T^{3} + 152377 p^{3} T^{4} + 248 p^{6} T^{5} + p^{9} T^{6}
47S4×C2S_4\times C_2 1−11pT+288700T2−106705481T3+288700p3T4−11p7T5+p9T6 1 - 11 p T + 288700 T^{2} - 106705481 T^{3} + 288700 p^{3} T^{4} - 11 p^{7} T^{5} + p^{9} T^{6}
53S4×C2S_4\times C_2 1−190T+285155T2−38423188T3+285155p3T4−190p6T5+p9T6 1 - 190 T + 285155 T^{2} - 38423188 T^{3} + 285155 p^{3} T^{4} - 190 p^{6} T^{5} + p^{9} T^{6}
59S4×C2S_4\times C_2 1+996T+829737T2+392514584T3+829737p3T4+996p6T5+p9T6 1 + 996 T + 829737 T^{2} + 392514584 T^{3} + 829737 p^{3} T^{4} + 996 p^{6} T^{5} + p^{9} T^{6}
61S4×C2S_4\times C_2 1−1214T+1069787T2−564015572T3+1069787p3T4−1214p6T5+p9T6 1 - 1214 T + 1069787 T^{2} - 564015572 T^{3} + 1069787 p^{3} T^{4} - 1214 p^{6} T^{5} + p^{9} T^{6}
67S4×C2S_4\times C_2 1−776T+970373T2−452560232T3+970373p3T4−776p6T5+p9T6 1 - 776 T + 970373 T^{2} - 452560232 T^{3} + 970373 p^{3} T^{4} - 776 p^{6} T^{5} + p^{9} T^{6}
71S4×C2S_4\times C_2 1−229T+917452T2−164982785T3+917452p3T4−229p6T5+p9T6 1 - 229 T + 917452 T^{2} - 164982785 T^{3} + 917452 p^{3} T^{4} - 229 p^{6} T^{5} + p^{9} T^{6}
73S4×C2S_4\times C_2 1+333T+1019422T2+267801129T3+1019422p3T4+333p6T5+p9T6 1 + 333 T + 1019422 T^{2} + 267801129 T^{3} + 1019422 p^{3} T^{4} + 333 p^{6} T^{5} + p^{9} T^{6}
79S4×C2S_4\times C_2 1+26T+1046041T2+20149636T3+1046041p3T4+26p6T5+p9T6 1 + 26 T + 1046041 T^{2} + 20149636 T^{3} + 1046041 p^{3} T^{4} + 26 p^{6} T^{5} + p^{9} T^{6}
83S4×C2S_4\times C_2 1−474T+1539841T2−550884324T3+1539841p3T4−474p6T5+p9T6 1 - 474 T + 1539841 T^{2} - 550884324 T^{3} + 1539841 p^{3} T^{4} - 474 p^{6} T^{5} + p^{9} T^{6}
89S4×C2S_4\times C_2 1+1320T+1590475T2+1206020752T3+1590475p3T4+1320p6T5+p9T6 1 + 1320 T + 1590475 T^{2} + 1206020752 T^{3} + 1590475 p^{3} T^{4} + 1320 p^{6} T^{5} + p^{9} T^{6}
97S4×C2S_4\times C_2 1−2386T+3755947T2−3905686572T3+3755947p3T4−2386p6T5+p9T6 1 - 2386 T + 3755947 T^{2} - 3905686572 T^{3} + 3755947 p^{3} T^{4} - 2386 p^{6} T^{5} + p^{9} T^{6}
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   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

−8.237425085007902047080101327433, −7.82199615019895581303515034606, −7.71955711647440220419497119394, −7.49958533632258940379596541152, −6.84745854138760102590960795811, −6.68529149947993891720702869995, −6.38515852651787941929973385490, −6.16466567356160506796398532773, −5.82203570554423947594225853644, −5.76438349573267361093507428935, −4.99913652983588003706459785330, −4.88289404396025635693092396452, −4.81091255969224530019384654138, −4.31959279849179437354974343203, −4.06957043727831851358021898624, −3.77778994704447044352732442110, −3.40605705360710064514985473825, −2.99684985243805051448208769333, −2.74641715741112312058985358223, −2.31350265054155103460987206531, −2.16880594163503479709839896277, −1.27723077123145478346334681322, −0.988955090345996178126488369965, −0.798678382865003062382215297000, −0.18693472396675260978476269932, 0.18693472396675260978476269932, 0.798678382865003062382215297000, 0.988955090345996178126488369965, 1.27723077123145478346334681322, 2.16880594163503479709839896277, 2.31350265054155103460987206531, 2.74641715741112312058985358223, 2.99684985243805051448208769333, 3.40605705360710064514985473825, 3.77778994704447044352732442110, 4.06957043727831851358021898624, 4.31959279849179437354974343203, 4.81091255969224530019384654138, 4.88289404396025635693092396452, 4.99913652983588003706459785330, 5.76438349573267361093507428935, 5.82203570554423947594225853644, 6.16466567356160506796398532773, 6.38515852651787941929973385490, 6.68529149947993891720702869995, 6.84745854138760102590960795811, 7.49958533632258940379596541152, 7.71955711647440220419497119394, 7.82199615019895581303515034606, 8.237425085007902047080101327433

Graph of the ZZ-function along the critical line