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Properties

Label 12-4840e6-1.1-c1e6-0-4
Degree $12$
Conductor $1.286\times 10^{22}$
Sign $1$
Analytic cond. $3.33222\times 10^{9}$
Root an. cond. $6.21671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $6$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 6·5-s − 4·7-s − 5·9-s − 12·15-s − 8·17-s − 12·19-s + 8·21-s − 8·23-s + 21·25-s + 10·27-s − 16·29-s − 4·31-s − 24·35-s + 8·37-s − 32·41-s + 4·43-s − 30·45-s − 6·47-s − 5·49-s + 16·51-s + 8·53-s + 24·57-s + 4·59-s − 16·61-s + 20·63-s − 2·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 2.68·5-s − 1.51·7-s − 5/3·9-s − 3.09·15-s − 1.94·17-s − 2.75·19-s + 1.74·21-s − 1.66·23-s + 21/5·25-s + 1.92·27-s − 2.97·29-s − 0.718·31-s − 4.05·35-s + 1.31·37-s − 4.99·41-s + 0.609·43-s − 4.47·45-s − 0.875·47-s − 5/7·49-s + 2.24·51-s + 1.09·53-s + 3.17·57-s + 0.520·59-s − 2.04·61-s + 2.51·63-s − 0.244·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{18} \cdot 5^{6} \cdot 11^{12}\)
Sign: $1$
Analytic conductor: \(3.33222\times 10^{9}\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(6\)
Selberg data: \((12,\ 2^{18} \cdot 5^{6} \cdot 11^{12} ,\ ( \ : [1/2]^{6} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( ( 1 - T )^{6} \)
11 \( 1 \)
good3 \( 1 + 2 T + p^{2} T^{2} + 2 p^{2} T^{3} + 50 T^{4} + 82 T^{5} + 181 T^{6} + 82 p T^{7} + 50 p^{2} T^{8} + 2 p^{5} T^{9} + p^{6} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
7 \( 1 + 4 T + 3 p T^{2} + 8 p T^{3} + 218 T^{4} + 636 T^{5} + 1993 T^{6} + 636 p T^{7} + 218 p^{2} T^{8} + 8 p^{4} T^{9} + 3 p^{5} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
13 \( 1 + 46 T^{2} + 16 T^{3} + 935 T^{4} + 560 T^{5} + 13172 T^{6} + 560 p T^{7} + 935 p^{2} T^{8} + 16 p^{3} T^{9} + 46 p^{4} T^{10} + p^{6} T^{12} \)
17 \( 1 + 8 T + 86 T^{2} + 488 T^{3} + 191 p T^{4} + 14352 T^{5} + 70772 T^{6} + 14352 p T^{7} + 191 p^{3} T^{8} + 488 p^{3} T^{9} + 86 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
19 \( 1 + 12 T + 142 T^{2} + 1028 T^{3} + 6983 T^{4} + 36232 T^{5} + 176372 T^{6} + 36232 p T^{7} + 6983 p^{2} T^{8} + 1028 p^{3} T^{9} + 142 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
23 \( 1 + 8 T + 78 T^{2} + 464 T^{3} + 3647 T^{4} + 17448 T^{5} + 99796 T^{6} + 17448 p T^{7} + 3647 p^{2} T^{8} + 464 p^{3} T^{9} + 78 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \)
29 \( 1 + 16 T + 158 T^{2} + 976 T^{3} + 3655 T^{4} + 5856 T^{5} - 10876 T^{6} + 5856 p T^{7} + 3655 p^{2} T^{8} + 976 p^{3} T^{9} + 158 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
31 \( 1 + 4 T + 106 T^{2} + 580 T^{3} + 5471 T^{4} + 35008 T^{5} + 194684 T^{6} + 35008 p T^{7} + 5471 p^{2} T^{8} + 580 p^{3} T^{9} + 106 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} \)
37 \( 1 - 8 T + 142 T^{2} - 872 T^{3} + 9911 T^{4} - 52016 T^{5} + 449252 T^{6} - 52016 p T^{7} + 9911 p^{2} T^{8} - 872 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
41 \( 1 + 32 T + 597 T^{2} + 7760 T^{3} + 78494 T^{4} + 647040 T^{5} + 4497049 T^{6} + 647040 p T^{7} + 78494 p^{2} T^{8} + 7760 p^{3} T^{9} + 597 p^{4} T^{10} + 32 p^{5} T^{11} + p^{6} T^{12} \)
43 \( 1 - 4 T + 157 T^{2} - 160 T^{3} + 9938 T^{4} + 11060 T^{5} + 442193 T^{6} + 11060 p T^{7} + 9938 p^{2} T^{8} - 160 p^{3} T^{9} + 157 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
47 \( 1 + 6 T + 217 T^{2} + 1142 T^{3} + 22146 T^{4} + 96502 T^{5} + 1330941 T^{6} + 96502 p T^{7} + 22146 p^{2} T^{8} + 1142 p^{3} T^{9} + 217 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} \)
53 \( 1 - 8 T + 142 T^{2} - 624 T^{3} + 8055 T^{4} - 33704 T^{5} + 443124 T^{6} - 33704 p T^{7} + 8055 p^{2} T^{8} - 624 p^{3} T^{9} + 142 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \)
59 \( 1 - 4 T + 106 T^{2} - 804 T^{3} + 11415 T^{4} - 59440 T^{5} + 798588 T^{6} - 59440 p T^{7} + 11415 p^{2} T^{8} - 804 p^{3} T^{9} + 106 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \)
61 \( 1 + 16 T + 245 T^{2} + 2240 T^{3} + 24374 T^{4} + 199632 T^{5} + 1856313 T^{6} + 199632 p T^{7} + 24374 p^{2} T^{8} + 2240 p^{3} T^{9} + 245 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
67 \( 1 + 2 T + 49 T^{2} + 458 T^{3} + 11018 T^{4} + 49970 T^{5} + 322277 T^{6} + 49970 p T^{7} + 11018 p^{2} T^{8} + 458 p^{3} T^{9} + 49 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} \)
71 \( 1 + 28 T + 546 T^{2} + 7900 T^{3} + 98591 T^{4} + 1012032 T^{5} + 9200236 T^{6} + 1012032 p T^{7} + 98591 p^{2} T^{8} + 7900 p^{3} T^{9} + 546 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} \)
73 \( 1 + 16 T + 246 T^{2} + 2768 T^{3} + 28991 T^{4} + 286368 T^{5} + 2601844 T^{6} + 286368 p T^{7} + 28991 p^{2} T^{8} + 2768 p^{3} T^{9} + 246 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} \)
79 \( 1 + 342 T^{2} + 288 T^{3} + 55791 T^{4} + 50976 T^{5} + 5551220 T^{6} + 50976 p T^{7} + 55791 p^{2} T^{8} + 288 p^{3} T^{9} + 342 p^{4} T^{10} + p^{6} T^{12} \)
83 \( 1 + 12 T + 210 T^{2} + 1028 T^{3} + 22311 T^{4} + 168072 T^{5} + 2881772 T^{6} + 168072 p T^{7} + 22311 p^{2} T^{8} + 1028 p^{3} T^{9} + 210 p^{4} T^{10} + 12 p^{5} T^{11} + p^{6} T^{12} \)
89 \( 1 - 18 T + 253 T^{2} - 1798 T^{3} + 8058 T^{4} + 72934 T^{5} - 1088091 T^{6} + 72934 p T^{7} + 8058 p^{2} T^{8} - 1798 p^{3} T^{9} + 253 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} \)
97 \( 1 + 342 T^{2} + 1080 T^{3} + 57231 T^{4} + 248616 T^{5} + 6595652 T^{6} + 248616 p T^{7} + 57231 p^{2} T^{8} + 1080 p^{3} T^{9} + 342 p^{4} T^{10} + p^{6} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.82940116053496188437579965137, −4.38697346658244414764206754907, −4.35028037059269154736930087422, −4.19144598231374951704796550641, −4.17402206086602341137051794799, −4.06553203345657894910602162357, −3.99280257085995928080592881090, −3.52062240393997614157804764761, −3.37222660974894868823018848867, −3.32921233561117657834995531154, −3.21061711648312574027150669985, −3.16405449445578732334569760020, −3.10575758315605535131095567112, −2.59117423742805720460575045348, −2.57125622261674315951456214531, −2.45450545892824961281589036281, −2.17253712569091146842021731432, −2.16757284309816279939303211315, −2.15175402636610473321615796557, −1.88672802065170928107240877298, −1.71710013420403844786507845850, −1.50195126474200778674830870961, −1.44676036649645738241952603334, −1.12300771710951077959255575604, −1.01155350655029705685685737547, 0, 0, 0, 0, 0, 0, 1.01155350655029705685685737547, 1.12300771710951077959255575604, 1.44676036649645738241952603334, 1.50195126474200778674830870961, 1.71710013420403844786507845850, 1.88672802065170928107240877298, 2.15175402636610473321615796557, 2.16757284309816279939303211315, 2.17253712569091146842021731432, 2.45450545892824961281589036281, 2.57125622261674315951456214531, 2.59117423742805720460575045348, 3.10575758315605535131095567112, 3.16405449445578732334569760020, 3.21061711648312574027150669985, 3.32921233561117657834995531154, 3.37222660974894868823018848867, 3.52062240393997614157804764761, 3.99280257085995928080592881090, 4.06553203345657894910602162357, 4.17402206086602341137051794799, 4.19144598231374951704796550641, 4.35028037059269154736930087422, 4.38697346658244414764206754907, 4.82940116053496188437579965137

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.