↑

Properties

Label 6-8512e3-1.1-c1e3-0-4
Degree 66
Conductor 616729673728616729673728
Sign −1-1
Analytic cond. 313997.313997.
Root an. cond. 8.244318.24431
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 33

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 2·5-s − 3·7-s + 9-s − 7·11-s + 2·13-s − 6·15-s + 7·17-s − 3·19-s + 9·21-s + 14·23-s − 6·25-s + 7·27-s + 3·29-s − 11·31-s + 21·33-s − 6·35-s − 6·39-s − 7·41-s + 4·43-s + 2·45-s + 8·47-s + 6·49-s − 21·51-s + 53-s − 14·55-s + 9·57-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.894·5-s − 1.13·7-s + 1/3·9-s − 2.11·11-s + 0.554·13-s − 1.54·15-s + 1.69·17-s − 0.688·19-s + 1.96·21-s + 2.91·23-s − 6/5·25-s + 1.34·27-s + 0.557·29-s − 1.97·31-s + 3.65·33-s − 1.01·35-s − 0.960·39-s − 1.09·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 6/7·49-s − 2.94·51-s + 0.137·53-s − 1.88·55-s + 1.19·57-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: −1-1
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: 33
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) == 00
L(12)L(\frac12) == 00
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+pT+8T2+14T3+8pT4+p3T5+p3T6 1 + p T + 8 T^{2} + 14 T^{3} + 8 p T^{4} + p^{3} T^{5} + p^{3} T^{6}
5S4×C2S_4\times C_2 1−2T+2pT2−18T3+2p2T4−2p2T5+p3T6 1 - 2 T + 2 p T^{2} - 18 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
11S4×C2S_4\times C_2 1+7T+4pT2+158T3+4p2T4+7p2T5+p3T6 1 + 7 T + 4 p T^{2} + 158 T^{3} + 4 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−2T+34T2−50T3+34pT4−2p2T5+p3T6 1 - 2 T + 34 T^{2} - 50 T^{3} + 34 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
17S4×C2S_4\times C_2 1−7T+40T2−132T3+40pT4−7p2T5+p3T6 1 - 7 T + 40 T^{2} - 132 T^{3} + 40 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1−14T+122T2−700T3+122pT4−14p2T5+p3T6 1 - 14 T + 122 T^{2} - 700 T^{3} + 122 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1−3T+14T2+104T3+14pT4−3p2T5+p3T6 1 - 3 T + 14 T^{2} + 104 T^{3} + 14 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1+11T+118T2+698T3+118pT4+11p2T5+p3T6 1 + 11 T + 118 T^{2} + 698 T^{3} + 118 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1+68T2−106T3+68pT4+p3T6 1 + 68 T^{2} - 106 T^{3} + 68 p T^{4} + p^{3} T^{6}
41S4×C2S_4\times C_2 1+7T−28T2−424T3−28pT4+7p2T5+p3T6 1 + 7 T - 28 T^{2} - 424 T^{3} - 28 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−4T+109T2−328T3+109pT4−4p2T5+p3T6 1 - 4 T + 109 T^{2} - 328 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1−8T+112T2−768T3+112pT4−8p2T5+p3T6 1 - 8 T + 112 T^{2} - 768 T^{3} + 112 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1−T+128T2−104T3+128pT4−p2T5+p3T6 1 - T + 128 T^{2} - 104 T^{3} + 128 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−10T+178T2−1056T3+178pT4−10p2T5+p3T6 1 - 10 T + 178 T^{2} - 1056 T^{3} + 178 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1−6T+134T2−650T3+134pT4−6p2T5+p3T6 1 - 6 T + 134 T^{2} - 650 T^{3} + 134 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−3T+122T2−214T3+122pT4−3p2T5+p3T6 1 - 3 T + 122 T^{2} - 214 T^{3} + 122 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+152T2−32T3+152pT4+p3T6 1 + 152 T^{2} - 32 T^{3} + 152 p T^{4} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−T+118T2−244T3+118pT4−p2T5+p3T6 1 - T + 118 T^{2} - 244 T^{3} + 118 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
79S4×C2S_4\times C_2 1+4T+193T2+664T3+193pT4+4p2T5+p3T6 1 + 4 T + 193 T^{2} + 664 T^{3} + 193 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6}
83S4×C2S_4\times C_2 1+31T+538T2+5934T3+538pT4+31p2T5+p3T6 1 + 31 T + 538 T^{2} + 5934 T^{3} + 538 p T^{4} + 31 p^{2} T^{5} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+28T+371T2+3632T3+371pT4+28p2T5+p3T6 1 + 28 T + 371 T^{2} + 3632 T^{3} + 371 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6}
97S4×C2S_4\times C_2 1+30T+534T2+6302T3+534pT4+30p2T5+p3T6 1 + 30 T + 534 T^{2} + 6302 T^{3} + 534 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6}
show more
show less
   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

−7.04784960504765531724880436671, −6.94213463281379938370616646878, −6.74015556110371023771053295938, −6.64351077901225897167722689450, −6.00618233763961226824705477669, −5.92812597983482153953793207022, −5.76791300880834702054866508529, −5.55690467379824512708195171073, −5.55564935763584390836687770564, −5.22371767619098008728360456176, −5.14100914060403736165816493811, −4.94436152225991249063541907881, −4.46571306500220432770236896418, −3.96348447361102782335170800345, −3.87729262164020350839383637659, −3.84074380917077995482881589127, −3.11297827182873428832739330777, −3.05691732576515462258334008097, −2.75063440134278432635368865278, −2.60550629150678625556847380484, −2.41380858491889938727993007836, −1.84872416717301476528816980793, −1.31803266948190901420974015913, −1.24742382205146565032243506579, −0.863009802359048103698523724741, 0, 0, 0, 0.863009802359048103698523724741, 1.24742382205146565032243506579, 1.31803266948190901420974015913, 1.84872416717301476528816980793, 2.41380858491889938727993007836, 2.60550629150678625556847380484, 2.75063440134278432635368865278, 3.05691732576515462258334008097, 3.11297827182873428832739330777, 3.84074380917077995482881589127, 3.87729262164020350839383637659, 3.96348447361102782335170800345, 4.46571306500220432770236896418, 4.94436152225991249063541907881, 5.14100914060403736165816493811, 5.22371767619098008728360456176, 5.55564935763584390836687770564, 5.55690467379824512708195171073, 5.76791300880834702054866508529, 5.92812597983482153953793207022, 6.00618233763961226824705477669, 6.64351077901225897167722689450, 6.74015556110371023771053295938, 6.94213463281379938370616646878, 7.04784960504765531724880436671

Graph of the ZZ-function along the critical line