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Properties

Label 6-9680e3-1.1-c1e3-0-7
Degree 66
Conductor 907039232000907039232000
Sign 11
Analytic cond. 461803.461803.
Root an. cond. 8.791768.79176
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·5-s − 7-s + 3·15-s − 7·17-s + 3·19-s − 21-s + 4·23-s + 6·25-s + 27-s − 5·29-s + 17·31-s − 3·35-s + 37-s + 2·41-s + 16·43-s + 6·47-s − 12·49-s − 7·51-s + 53-s + 3·57-s + 4·59-s − 11·61-s + 16·67-s + 4·69-s − 71-s + 22·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s − 0.377·7-s + 0.774·15-s − 1.69·17-s + 0.688·19-s − 0.218·21-s + 0.834·23-s + 6/5·25-s + 0.192·27-s − 0.928·29-s + 3.05·31-s − 0.507·35-s + 0.164·37-s + 0.312·41-s + 2.43·43-s + 0.875·47-s − 1.71·49-s − 0.980·51-s + 0.137·53-s + 0.397·57-s + 0.520·59-s − 1.40·61-s + 1.95·67-s + 0.481·69-s − 0.118·71-s + 2.57·73-s + ⋯

Functional equation

Λ(s)=((212⋅53⋅116)s/2 ΓC(s)3 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((212⋅53⋅116)s/2 ΓC(s+1/2)3 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \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: 212⋅53⋅1162^{12} \cdot 5^{3} \cdot 11^{6}
Sign: 11
Analytic conductor: 461803.461803.
Root analytic conductor: 8.791768.79176
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (6, 212⋅53⋅116, ( :1/2,1/2,1/2), 1)(6,\ 2^{12} \cdot 5^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 10.9214421010.92144210
L(12)L(\frac12) ≈\approx 10.9214421010.92144210
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}
11 1 1
good3S4×C2S_4\times C_2 1−T+T2−2T3+pT4−p2T5+p3T6 1 - T + T^{2} - 2 T^{3} + p T^{4} - p^{2} T^{5} + p^{3} T^{6}
7S4×C2S_4\times C_2 1+T+13T2+10T3+13pT4+p2T5+p3T6 1 + T + 13 T^{2} + 10 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
13S4×C2S_4\times C_2 1−T2−64T3−pT4+p3T6 1 - T^{2} - 64 T^{3} - p T^{4} + p^{3} T^{6}
17S4×C2S_4\times C_2 1+7T+59T2+230T3+59pT4+7p2T5+p3T6 1 + 7 T + 59 T^{2} + 230 T^{3} + 59 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6}
19S4×C2S_4\times C_2 1−3T+17T2−130T3+17pT4−3p2T5+p3T6 1 - 3 T + 17 T^{2} - 130 T^{3} + 17 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6}
23S4×C2S_4\times C_2 1−4T+49T2−120T3+49pT4−4p2T5+p3T6 1 - 4 T + 49 T^{2} - 120 T^{3} + 49 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
29S4×C2S_4\times C_2 1+5T+71T2+226T3+71pT4+5p2T5+p3T6 1 + 5 T + 71 T^{2} + 226 T^{3} + 71 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6}
31S4×C2S_4\times C_2 1−17T+165T2−1070T3+165pT4−17p2T5+p3T6 1 - 17 T + 165 T^{2} - 1070 T^{3} + 165 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6}
37S4×C2S_4\times C_2 1−T+87T2−94T3+87pT4−p2T5+p3T6 1 - T + 87 T^{2} - 94 T^{3} + 87 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
41S4×C2S_4\times C_2 1−2T+91T2−132T3+91pT4−2p2T5+p3T6 1 - 2 T + 91 T^{2} - 132 T^{3} + 91 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6}
43S4×C2S_4\times C_2 1−16T+189T2−1360T3+189pT4−16p2T5+p3T6 1 - 16 T + 189 T^{2} - 1360 T^{3} + 189 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6}
47S4×C2S_4\times C_2 1−6T+113T2−556T3+113pT4−6p2T5+p3T6 1 - 6 T + 113 T^{2} - 556 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6}
53S4×C2S_4\times C_2 1−T+135T2−126T3+135pT4−p2T5+p3T6 1 - T + 135 T^{2} - 126 T^{3} + 135 p T^{4} - p^{2} T^{5} + p^{3} T^{6}
59S4×C2S_4\times C_2 1−4T+81T2−600T3+81pT4−4p2T5+p3T6 1 - 4 T + 81 T^{2} - 600 T^{3} + 81 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6}
61S4×C2S_4\times C_2 1+11T+95T2+6pT3+95pT4+11p2T5+p3T6 1 + 11 T + 95 T^{2} + 6 p T^{3} + 95 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6}
67S4×C2S_4\times C_2 1−16T+165T2−1168T3+165pT4−16p2T5+p3T6 1 - 16 T + 165 T^{2} - 1168 T^{3} + 165 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6}
71S4×C2S_4\times C_2 1+T+85T2+654T3+85pT4+p2T5+p3T6 1 + T + 85 T^{2} + 654 T^{3} + 85 p T^{4} + p^{2} T^{5} + p^{3} T^{6}
73S4×C2S_4\times C_2 1−22T+355T2−3388T3+355pT4−22p2T5+p3T6 1 - 22 T + 355 T^{2} - 3388 T^{3} + 355 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6}
79C2C_2 (1−12T+pT2)3 ( 1 - 12 T + p T^{2} )^{3}
83S4×C2S_4\times C_2 1+21T2−880T3+21pT4+p3T6 1 + 21 T^{2} - 880 T^{3} + 21 p T^{4} + p^{3} T^{6}
89S4×C2S_4\times C_2 1+3T+83T2+1138T3+83pT4+3p2T5+p3T6 1 + 3 T + 83 T^{2} + 1138 T^{3} + 83 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6}
97C2C_2 (1−6T+pT2)3 ( 1 - 6 T + p T^{2} )^{3}
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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

−6.81536576017082901890951337774, −6.43364966222057861442194262641, −6.37546742156865966344056596767, −6.18264682903485965925015072763, −5.94311100845211582892545258324, −5.63778862520529089286875274799, −5.51064038249860586115086560871, −5.02041074488077072463101863985, −4.93426243357501218821942298246, −4.66605153796993040813196465347, −4.58978218514600644447283448560, −4.25326122194598661344793578469, −3.92739205104372921978307207270, −3.49091796461213655051491995745, −3.33206338406025818099414574435, −3.31262627352069103180232129545, −2.70940096289806897509148276314, −2.48867473817428016200642733849, −2.37912396688920701217631372703, −2.13366419772419433811967890920, −1.96725921779954477840261696818, −1.37158493718043754183518143793, −0.976010292333823414274707367195, −0.76643553671172750274623175089, −0.50923874372048305079146366620, 0.50923874372048305079146366620, 0.76643553671172750274623175089, 0.976010292333823414274707367195, 1.37158493718043754183518143793, 1.96725921779954477840261696818, 2.13366419772419433811967890920, 2.37912396688920701217631372703, 2.48867473817428016200642733849, 2.70940096289806897509148276314, 3.31262627352069103180232129545, 3.33206338406025818099414574435, 3.49091796461213655051491995745, 3.92739205104372921978307207270, 4.25326122194598661344793578469, 4.58978218514600644447283448560, 4.66605153796993040813196465347, 4.93426243357501218821942298246, 5.02041074488077072463101863985, 5.51064038249860586115086560871, 5.63778862520529089286875274799, 5.94311100845211582892545258324, 6.18264682903485965925015072763, 6.37546742156865966344056596767, 6.43364966222057861442194262641, 6.81536576017082901890951337774

Graph of the ZZ-function along the critical line