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Properties

Label 6-8512e3-1.1-c1e3-0-2
Degree 66
Conductor 616729673728616729673728
Sign 11
Analytic cond. 313997.313997.
Root an. cond. 8.244318.24431
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-s − 2·5-s − 3·7-s + 9-s + 7·11-s + 2·15-s + 17-s − 3·19-s + 3·21-s + 16·23-s − 2·25-s + 5·27-s + 29-s − 5·31-s − 7·33-s + 6·35-s + 6·37-s + 41-s − 4·43-s − 2·45-s − 14·47-s + 6·49-s − 51-s − 31·53-s − 14·55-s + 3·57-s + 18·59-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 1.13·7-s + 1/3·9-s + 2.11·11-s + 0.516·15-s + 0.242·17-s − 0.688·19-s + 0.654·21-s + 3.33·23-s − 2/5·25-s + 0.962·27-s + 0.185·29-s − 0.898·31-s − 1.21·33-s + 1.01·35-s + 0.986·37-s + 0.156·41-s − 0.609·43-s − 0.298·45-s − 2.04·47-s + 6/7·49-s − 0.140·51-s − 4.25·53-s − 1.88·55-s + 0.397·57-s + 2.34·59-s + ⋯

Functional equation

Λ(s)=((218⋅73⋅193)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((218⋅73⋅193)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 218⋅73⋅1932^{18} \cdot 7^{3} \cdot 19^{3}
Sign: 11
Analytic conductor: 313997.313997.
Root analytic conductor: 8.244318.24431
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 218⋅73⋅193, ( :1/2,1/2,1/2), 1)(6,\ 2^{18} \cdot 7^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.2841459542.284145954
L(12)L(\frac12) ≈\approx 2.2841459542.284145954
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
7C1C_1 (1+T)3 ( 1 + T )^{3}
19C1C_1 (1+T)3 ( 1 + T )^{3}
good3S4×C2S_4\times C_2 1+T−2pT3+p2T5+p3T6 1 + T - 2 p T^{3} + p^{2} T^{5} + p^{3} T^{6}
5S4×C2S_4\times C_2 1+2T+6T2+14T3+6pT4+2p2T5+p3T6 1 + 2 T + 6 T^{2} + 14 T^{3} + 6 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1−7T+34T2−118T3+34pT4−7p2T5+p3T6 1 - 7 T + 34 T^{2} - 118 T^{3} + 34 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}
13D6D_{6} 1−54T3+p3T6 1 - 54 T^{3} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−T+36T2−16T3+36pT4−p2T5+p3T6 1 - T + 36 T^{2} - 16 T^{3} + 36 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1−16T+144T2−832T3+144pT4−16p2T5+p3T6 1 - 16 T + 144 T^{2} - 832 T^{3} + 144 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1−T+72T2−40T3+72pT4−p2T5+p3T6 1 - T + 72 T^{2} - 40 T^{3} + 72 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1+5T+86T2+302T3+86pT4+5p2T5+p3T6 1 + 5 T + 86 T^{2} + 302 T^{3} + 86 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1−6T+30T2−282T3+30pT4−6p2T5+p3T6 1 - 6 T + 30 T^{2} - 282 T^{3} + 30 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1−T+42T2+200T3+42pT4−p2T5+p3T6 1 - T + 42 T^{2} + 200 T^{3} + 42 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1+4T+93T2+296T3+93pT4+4p2T5+p3T6 1 + 4 T + 93 T^{2} + 296 T^{3} + 93 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1+14T+150T2+1100T3+150pT4+14p2T5+p3T6 1 + 14 T + 150 T^{2} + 1100 T^{3} + 150 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+31T+470T2+4284T3+470pT4+31p2T5+p3T6 1 + 31 T + 470 T^{2} + 4284 T^{3} + 470 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−18T+246T2−2160T3+246pT4−18p2T5+p3T6 1 - 18 T + 246 T^{2} - 2160 T^{3} + 246 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1+26T+398T2+3734T3+398pT4+26p2T5+p3T6 1 + 26 T + 398 T^{2} + 3734 T^{3} + 398 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1+T+94T2+110T3+94pT4+p2T5+p3T6 1 + T + 94 T^{2} + 110 T^{3} + 94 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1−12T+168T2−1380T3+168pT4−12p2T5+p3T6 1 - 12 T + 168 T^{2} - 1380 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−19T+330T2−2960T3+330pT4−19p2T5+p3T6 1 - 19 T + 330 T^{2} - 2960 T^{3} + 330 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1+4T+201T2+584T3+201pT4+4p2T5+p3T6 1 + 4 T + 201 T^{2} + 584 T^{3} + 201 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1+13T+4T2−1070T3+4pT4+13p2T5+p3T6 1 + 13 T + 4 T^{2} - 1070 T^{3} + 4 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+12T+147T2+704T3+147pT4+12p2T5+p3T6 1 + 12 T + 147 T^{2} + 704 T^{3} + 147 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+8T+156T2+1906T3+156pT4+8p2T5+p3T6 1 + 8 T + 156 T^{2} + 1906 T^{3} + 156 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

−6.91642147420745210133188146587, −6.62184419204002041063340920303, −6.52336016123654829274922242743, −6.34052664001993887919436478307, −5.94184918207743366799289190590, −5.77540459115707175287224979171, −5.65637010065159159267056683458, −5.11382744227814027892088936851, −4.86180031944613324352861532458, −4.65117923035807252008061452378, −4.64520891481843036013958086628, −4.38123619603899161848870084130, −3.87311867729149724299568554032, −3.73640320085587360343548950637, −3.47452596270728101377169845472, −3.31485518232009150812517471510, −2.96045661514207416379855030556, −2.86258683528650375381443170712, −2.52825694833560734923918330967, −1.87072492040395193758747388314, −1.62713835083457126486192726010, −1.25016092033467334981864045882, −1.22135659841066034019150866158, −0.44431870555185554149840921430, −0.43498480492521483710492847745, 0.43498480492521483710492847745, 0.44431870555185554149840921430, 1.22135659841066034019150866158, 1.25016092033467334981864045882, 1.62713835083457126486192726010, 1.87072492040395193758747388314, 2.52825694833560734923918330967, 2.86258683528650375381443170712, 2.96045661514207416379855030556, 3.31485518232009150812517471510, 3.47452596270728101377169845472, 3.73640320085587360343548950637, 3.87311867729149724299568554032, 4.38123619603899161848870084130, 4.64520891481843036013958086628, 4.65117923035807252008061452378, 4.86180031944613324352861532458, 5.11382744227814027892088936851, 5.65637010065159159267056683458, 5.77540459115707175287224979171, 5.94184918207743366799289190590, 6.34052664001993887919436478307, 6.52336016123654829274922242743, 6.62184419204002041063340920303, 6.91642147420745210133188146587

Graph of the ZZ-function along the critical line