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Properties

Label 6-7360e3-1.1-c1e3-0-3
Degree 66
Conductor 398688256000398688256000
Sign 11
Analytic cond. 202985.202985.
Root an. cond. 7.666157.66615
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 − 3·5-s − 2·7-s + 9-s + 7·11-s + 13-s − 3·15-s + 10·17-s − 13·19-s − 2·21-s − 3·23-s + 6·25-s − 27-s − 13·29-s + 8·31-s + 7·33-s + 6·35-s − 5·37-s + 39-s + 8·41-s + 24·43-s − 3·45-s − 2·47-s − 9·49-s + 10·51-s − 53-s − 21·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 0.755·7-s + 1/3·9-s + 2.11·11-s + 0.277·13-s − 0.774·15-s + 2.42·17-s − 2.98·19-s − 0.436·21-s − 0.625·23-s + 6/5·25-s − 0.192·27-s − 2.41·29-s + 1.43·31-s + 1.21·33-s + 1.01·35-s − 0.821·37-s + 0.160·39-s + 1.24·41-s + 3.65·43-s − 0.447·45-s − 0.291·47-s − 9/7·49-s + 1.40·51-s − 0.137·53-s − 2.83·55-s + ⋯

Functional equation

Λ(s)=((218⋅53⋅233)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((218⋅53⋅233)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 218⋅53⋅2332^{18} \cdot 5^{3} \cdot 23^{3}
Sign: 11
Analytic conductor: 202985.202985.
Root analytic conductor: 7.666157.66615
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 218⋅53⋅233, ( :1/2,1/2,1/2), 1)(6,\ 2^{18} \cdot 5^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 2.7204884212.720488421
L(12)L(\frac12) ≈\approx 2.7204884212.720488421
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
5C1C_1 (1+T)3 ( 1 + T )^{3}
23C1C_1 (1+T)3 ( 1 + T )^{3}
good3S4×C2S_4\times C_2 1−T+2T3−p2T5+p3T6 1 - T + 2 T^{3} - p^{2} T^{5} + p^{3} T^{6}
7S4×C2S_4\times C_2 1+2T+13T2+27T3+13pT4+2p2T5+p3T6 1 + 2 T + 13 T^{2} + 27 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1−7T+40T2−140T3+40pT4−7p2T5+p3T6 1 - 7 T + 40 T^{2} - 140 T^{3} + 40 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−T+16T2+24T3+16pT4−p2T5+p3T6 1 - T + 16 T^{2} + 24 T^{3} + 16 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−10T+61T2−257T3+61pT4−10p2T5+p3T6 1 - 10 T + 61 T^{2} - 257 T^{3} + 61 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}
19S4×C2S_4\times C_2 1+13T+90T2+432T3+90pT4+13p2T5+p3T6 1 + 13 T + 90 T^{2} + 432 T^{3} + 90 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1+13T+113T2+678T3+113pT4+13p2T5+p3T6 1 + 13 T + 113 T^{2} + 678 T^{3} + 113 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−8T+105T2−495T3+105pT4−8p2T5+p3T6 1 - 8 T + 105 T^{2} - 495 T^{3} + 105 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+5T+19T2−126T3+19pT4+5p2T5+p3T6 1 + 5 T + 19 T^{2} - 126 T^{3} + 19 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1−8T+121T2−655T3+121pT4−8p2T5+p3T6 1 - 8 T + 121 T^{2} - 655 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
43C2C_2 (1−8T+pT2)3 ( 1 - 8 T + p T^{2} )^{3}
47S4×C2S_4\times C_2 1+2T+49T2−212T3+49pT4+2p2T5+p3T6 1 + 2 T + 49 T^{2} - 212 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1+T+59T2+406T3+59pT4+p2T5+p3T6 1 + T + 59 T^{2} + 406 T^{3} + 59 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1+17T+243T2+2042T3+243pT4+17p2T5+p3T6 1 + 17 T + 243 T^{2} + 2042 T^{3} + 243 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1+13T+132T2+866T3+132pT4+13p2T5+p3T6 1 + 13 T + 132 T^{2} + 866 T^{3} + 132 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−5T+109T2−174T3+109pT4−5p2T5+p3T6 1 - 5 T + 109 T^{2} - 174 T^{3} + 109 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1−22T+309T2−2899T3+309pT4−22p2T5+p3T6 1 - 22 T + 309 T^{2} - 2899 T^{3} + 309 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−12T+43T2+368T3+43pT4−12p2T5+p3T6 1 - 12 T + 43 T^{2} + 368 T^{3} + 43 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1−4T+93T2−120T3+93pT4−4p2T5+p3T6 1 - 4 T + 93 T^{2} - 120 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1+15T+289T2+2454T3+289pT4+15p2T5+p3T6 1 + 15 T + 289 T^{2} + 2454 T^{3} + 289 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+8T+139T2+1360T3+139pT4+8p2T5+p3T6 1 + 8 T + 139 T^{2} + 1360 T^{3} + 139 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+3T+210T2+632T3+210pT4+3p2T5+p3T6 1 + 3 T + 210 T^{2} + 632 T^{3} + 210 p T^{4} + 3 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

−6.81958857123440871293661366130, −6.65347211365554599901387542844, −6.56611384038250505653043893596, −6.54071180711964683928202502563, −6.06436684263477582493301994571, −5.77513251299169465228876921778, −5.74625787438582483704658803682, −5.40092164812512944042898637357, −5.10848134340511699269751344980, −4.62132126163058058408218737972, −4.29633131407904759341016497101, −4.17071427905431246841868577944, −4.13622685959038549297121858750, −3.77048350142978692165784547517, −3.63097656764416510019899283663, −3.53673367842698188455855425497, −2.85005657271218686986235037289, −2.84938167841183354427928620375, −2.62619672697677190812160725695, −2.04598562968674211633358081153, −1.71884993646833810372214609288, −1.41722309508015648566065006397, −1.24340931689433925516283706346, −0.58087134435570794693660983215, −0.33867661168378627352240419280, 0.33867661168378627352240419280, 0.58087134435570794693660983215, 1.24340931689433925516283706346, 1.41722309508015648566065006397, 1.71884993646833810372214609288, 2.04598562968674211633358081153, 2.62619672697677190812160725695, 2.84938167841183354427928620375, 2.85005657271218686986235037289, 3.53673367842698188455855425497, 3.63097656764416510019899283663, 3.77048350142978692165784547517, 4.13622685959038549297121858750, 4.17071427905431246841868577944, 4.29633131407904759341016497101, 4.62132126163058058408218737972, 5.10848134340511699269751344980, 5.40092164812512944042898637357, 5.74625787438582483704658803682, 5.77513251299169465228876921778, 6.06436684263477582493301994571, 6.54071180711964683928202502563, 6.56611384038250505653043893596, 6.65347211365554599901387542844, 6.81958857123440871293661366130

Graph of the ZZ-function along the critical line