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Properties

Label 6-9075e3-1.1-c1e3-0-7
Degree 66
Conductor 747377296875747377296875
Sign 11
Analytic cond. 380514.380514.
Root an. cond. 8.512598.51259
Motivic weight 11
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  + 3·2-s + 3·3-s + 4·4-s + 9·6-s + 8·7-s + 4·8-s + 6·9-s + 12·12-s + 6·13-s + 24·14-s + 3·16-s + 4·17-s + 18·18-s + 2·19-s + 24·21-s − 12·23-s + 12·24-s + 18·26-s + 10·27-s + 32·28-s + 8·29-s + 8·31-s − 32-s + 12·34-s + 24·36-s − 4·37-s + 6·38-s + ⋯
L(s)  = 1  + 2.12·2-s + 1.73·3-s + 2·4-s + 3.67·6-s + 3.02·7-s + 1.41·8-s + 2·9-s + 3.46·12-s + 1.66·13-s + 6.41·14-s + 3/4·16-s + 0.970·17-s + 4.24·18-s + 0.458·19-s + 5.23·21-s − 2.50·23-s + 2.44·24-s + 3.53·26-s + 1.92·27-s + 6.04·28-s + 1.48·29-s + 1.43·31-s − 0.176·32-s + 2.05·34-s + 4·36-s − 0.657·37-s + 0.973·38-s + ⋯

Functional equation

Λ(s)=((33⋅56⋅116)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((33⋅56⋅116)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 33⋅56⋅1163^{3} \cdot 5^{6} \cdot 11^{6}
Sign: 11
Analytic conductor: 380514.380514.
Root analytic conductor: 8.512598.51259
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 33⋅56⋅116, ( :1/2,1/2,1/2), 1)(6,\ 3^{3} \cdot 5^{6} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 104.0981210104.0981210
L(12)L(\frac12) ≈\approx 104.0981210104.0981210
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)
bad3C1C_1 (1−T)3 ( 1 - T )^{3}
5 1 1
11 1 1
good2S4×C2S_4\times C_2 1−3T+5T2−7T3+5pT4−3p2T5+p3T6 1 - 3 T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
7S4×C2S_4\times C_2 1−8T+37T2−116T3+37pT4−8p2T5+p3T6 1 - 8 T + 37 T^{2} - 116 T^{3} + 37 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−6T+11T2−8T3+11pT4−6p2T5+p3T6 1 - 6 T + 11 T^{2} - 8 T^{3} + 11 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−4T+23T2−20T3+23pT4−4p2T5+p3T6 1 - 4 T + 23 T^{2} - 20 T^{3} + 23 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
19S4×C2S_4\times C_2 1−2T+5T2+108T3+5pT4−2p2T5+p3T6 1 - 2 T + 5 T^{2} + 108 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
23C2C_2 (1+4T+pT2)3 ( 1 + 4 T + p T^{2} )^{3}
29S4×C2S_4\times C_2 1−8T+3pT2−432T3+3p2T4−8p2T5+p3T6 1 - 8 T + 3 p T^{2} - 432 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−8T+101T2−480T3+101pT4−8p2T5+p3T6 1 - 8 T + 101 T^{2} - 480 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+4T+95T2+264T3+95pT4+4p2T5+p3T6 1 + 4 T + 95 T^{2} + 264 T^{3} + 95 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1−8T+3pT2−624T3+3p2T4−8p2T5+p3T6 1 - 8 T + 3 p T^{2} - 624 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−8T+145T2−692T3+145pT4−8p2T5+p3T6 1 - 8 T + 145 T^{2} - 692 T^{3} + 145 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1+8T+125T2+592T3+125pT4+8p2T5+p3T6 1 + 8 T + 125 T^{2} + 592 T^{3} + 125 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1−8T+127T2−576T3+127pT4−8p2T5+p3T6 1 - 8 T + 127 T^{2} - 576 T^{3} + 127 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1+8T+113T2+1024T3+113pT4+8p2T5+p3T6 1 + 8 T + 113 T^{2} + 1024 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1+2T+131T2+204T3+131pT4+2p2T5+p3T6 1 + 2 T + 131 T^{2} + 204 T^{3} + 131 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1+12T+185T2+1288T3+185pT4+12p2T5+p3T6 1 + 12 T + 185 T^{2} + 1288 T^{3} + 185 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+12T+245T2+1688T3+245pT4+12p2T5+p3T6 1 + 12 T + 245 T^{2} + 1688 T^{3} + 245 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−18T+279T2−2536T3+279pT4−18p2T5+p3T6 1 - 18 T + 279 T^{2} - 2536 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1−6T+233T2−940T3+233pT4−6p2T5+p3T6 1 - 6 T + 233 T^{2} - 940 T^{3} + 233 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1+2T+245T2+328T3+245pT4+2p2T5+p3T6 1 + 2 T + 245 T^{2} + 328 T^{3} + 245 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1−2T+255T2−348T3+255pT4−2p2T5+p3T6 1 - 2 T + 255 T^{2} - 348 T^{3} + 255 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1−8T+259T2−1424T3+259pT4−8p2T5+p3T6 1 - 8 T + 259 T^{2} - 1424 T^{3} + 259 p T^{4} - 8 p^{2} T^{5} + p^{3} 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

−6.72176784957749855626729729711, −6.61325391178579816190892295001, −6.15729711952447827858920300835, −6.04263812970950156136323919279, −5.82211335118011540854835073926, −5.51720600073119788253420642123, −5.50010325320873716703442156154, −4.88869924655958038353473046937, −4.79002671290631610833740203557, −4.63578581259931773563150557616, −4.41398244738947921079902727578, −4.29159704524203784988612238903, −4.22852520169071384707905396936, −3.58088400067104101410860796764, −3.51000449289901905732841120458, −3.48285827296203539637603309202, −2.96850132261302279008297676182, −2.86039017852520544331053773583, −2.21355586031351909025284776293, −2.12674544853775588197824821184, −1.97671941444564389074796256942, −1.64638984761595203001268343530, −1.32750749544657828394095531713, −0.873204115582791448545432409592, −0.833837575694138690640980516308, 0.833837575694138690640980516308, 0.873204115582791448545432409592, 1.32750749544657828394095531713, 1.64638984761595203001268343530, 1.97671941444564389074796256942, 2.12674544853775588197824821184, 2.21355586031351909025284776293, 2.86039017852520544331053773583, 2.96850132261302279008297676182, 3.48285827296203539637603309202, 3.51000449289901905732841120458, 3.58088400067104101410860796764, 4.22852520169071384707905396936, 4.29159704524203784988612238903, 4.41398244738947921079902727578, 4.63578581259931773563150557616, 4.79002671290631610833740203557, 4.88869924655958038353473046937, 5.50010325320873716703442156154, 5.51720600073119788253420642123, 5.82211335118011540854835073926, 6.04263812970950156136323919279, 6.15729711952447827858920300835, 6.61325391178579816190892295001, 6.72176784957749855626729729711

Graph of the ZZ-function along the critical line