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Properties

Label 6-525e3-1.1-c1e3-0-0
Degree 66
Conductor 144703125144703125
Sign 11
Analytic cond. 73.673173.6731
Root an. cond. 2.047472.04747
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  + 2-s − 3·3-s − 3·6-s − 3·7-s + 2·8-s + 6·9-s + 6·11-s − 6·13-s − 3·14-s + 3·16-s + 6·18-s + 6·19-s + 9·21-s + 6·22-s + 4·23-s − 6·24-s − 6·26-s − 10·27-s + 2·29-s + 2·31-s + 3·32-s − 18·33-s − 4·37-s + 6·38-s + 18·39-s + 2·41-s + 9·42-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s − 1.22·6-s − 1.13·7-s + 0.707·8-s + 2·9-s + 1.80·11-s − 1.66·13-s − 0.801·14-s + 3/4·16-s + 1.41·18-s + 1.37·19-s + 1.96·21-s + 1.27·22-s + 0.834·23-s − 1.22·24-s − 1.17·26-s − 1.92·27-s + 0.371·29-s + 0.359·31-s + 0.530·32-s − 3.13·33-s − 0.657·37-s + 0.973·38-s + 2.88·39-s + 0.312·41-s + 1.38·42-s + ⋯

Functional equation

Λ(s)=((33⋅56⋅73)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((33⋅56⋅73)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 66
Conductor: 33⋅56⋅733^{3} \cdot 5^{6} \cdot 7^{3}
Sign: 11
Analytic conductor: 73.673173.6731
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 33⋅56⋅73, ( :1/2,1/2,1/2), 1)(6,\ 3^{3} \cdot 5^{6} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.8579302011.857930201
L(12)L(\frac12) ≈\approx 1.8579302011.857930201
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
7C1C_1 (1+T)3 ( 1 + T )^{3}
good2S4×C2S_4\times C_2 1−T+T2−3T3+pT4−p2T5+p3T6 1 - T + T^{2} - 3 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6}
11C2C_2 (1−2T+pT2)3 ( 1 - 2 T + p T^{2} )^{3}
13S4×C2S_4\times C_2 1+6T+35T2+148T3+35pT4+6p2T5+p3T6 1 + 6 T + 35 T^{2} + 148 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1+35T2−16T3+35pT4+p3T6 1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6}
19S4×C2S_4\times C_2 1−6T+53T2−188T3+53pT4−6p2T5+p3T6 1 - 6 T + 53 T^{2} - 188 T^{3} + 53 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1−4T+61T2−168T3+61pT4−4p2T5+p3T6 1 - 4 T + 61 T^{2} - 168 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1−2T+35T2−76T3+35pT4−2p2T5+p3T6 1 - 2 T + 35 T^{2} - 76 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−2T+41T2+60T3+41pT4−2p2T5+p3T6 1 - 2 T + 41 T^{2} + 60 T^{3} + 41 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+4T+31T2+232T3+31pT4+4p2T5+p3T6 1 + 4 T + 31 T^{2} + 232 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1−2T+63T2+36T3+63pT4−2p2T5+p3T6 1 - 2 T + 63 T^{2} + 36 T^{3} + 63 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−4T−15T2+488T3−15pT4−4p2T5+p3T6 1 - 4 T - 15 T^{2} + 488 T^{3} - 15 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1−8T+109T2−624T3+109pT4−8p2T5+p3T6 1 - 8 T + 109 T^{2} - 624 T^{3} + 109 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1−14T+171T2−1188T3+171pT4−14p2T5+p3T6 1 - 14 T + 171 T^{2} - 1188 T^{3} + 171 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−16T+113T2−608T3+113pT4−16p2T5+p3T6 1 - 16 T + 113 T^{2} - 608 T^{3} + 113 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1+6T+131T2+484T3+131pT4+6p2T5+p3T6 1 + 6 T + 131 T^{2} + 484 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−8T+169T2−944T3+169pT4−8p2T5+p3T6 1 - 8 T + 169 T^{2} - 944 T^{3} + 169 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
71C2C_2 (1−2T+pT2)3 ( 1 - 2 T + p T^{2} )^{3}
73S4×C2S_4\times C_2 1+18T+311T2+2732T3+311pT4+18p2T5+p3T6 1 + 18 T + 311 T^{2} + 2732 T^{3} + 311 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1−12T+221T2−1576T3+221pT4−12p2T5+p3T6 1 - 12 T + 221 T^{2} - 1576 T^{3} + 221 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1−8T+185T2−1072T3+185pT4−8p2T5+p3T6 1 - 8 T + 185 T^{2} - 1072 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1−14T+319T2−2532T3+319pT4−14p2T5+p3T6 1 - 14 T + 319 T^{2} - 2532 T^{3} + 319 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+22T+255T2+2404T3+255pT4+22p2T5+p3T6 1 + 22 T + 255 T^{2} + 2404 T^{3} + 255 p T^{4} + 22 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

−9.914525071408972659806251594373, −9.316520234174628371140068914352, −9.284020288732556461940958544307, −9.156825295931349904739131572860, −8.362766972126603023432502689627, −8.230490685348470661746481354447, −7.54195956443254506082870524233, −7.33870709268040940690541670912, −7.06666112289600058325462250884, −6.87566073734968804379748228912, −6.64213211134235603508882279981, −6.18421278721861225303735663511, −5.92690994891109039255491340178, −5.34873866904477615142468144811, −5.27741902028662012725921251974, −5.14085153600714544753454493451, −4.30027321369541196988573082776, −4.22140353570702312656420354378, −4.17650345759519333851478902367, −3.35778747230251275316332301302, −3.14182764210480382492854285569, −2.43619215235081207014809590862, −1.89682845780111182442129770040, −0.956151394633825783233879657697, −0.799153297890762279963436329692, 0.799153297890762279963436329692, 0.956151394633825783233879657697, 1.89682845780111182442129770040, 2.43619215235081207014809590862, 3.14182764210480382492854285569, 3.35778747230251275316332301302, 4.17650345759519333851478902367, 4.22140353570702312656420354378, 4.30027321369541196988573082776, 5.14085153600714544753454493451, 5.27741902028662012725921251974, 5.34873866904477615142468144811, 5.92690994891109039255491340178, 6.18421278721861225303735663511, 6.64213211134235603508882279981, 6.87566073734968804379748228912, 7.06666112289600058325462250884, 7.33870709268040940690541670912, 7.54195956443254506082870524233, 8.230490685348470661746481354447, 8.362766972126603023432502689627, 9.156825295931349904739131572860, 9.284020288732556461940958544307, 9.316520234174628371140068914352, 9.914525071408972659806251594373

Graph of the ZZ-function along the critical line