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Degree Bad \(p\) Root analytic conductor Motivic weight Spectral label
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Columns to display Euler factor constraints
e.g. 3,7,19 e.g. F3=1-T,F5=1+T+5T^2

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  displayed columns for results
$L(s) = \prod_p F_p(p^{-s})^{-1}$
Label Origin \(F_{ 2 }(T)\) \(F_{ 3 }(T)\) \(F_{ 5 }(T)\) \(F_{ 7 }(T)\) \(F_{ 11 }(T)\) \(F_{ 13 }(T)\) \(F_{ 17 }(T)\) \(F_{ 19 }(T)\) \(F_{ 23 }(T)\) \(F_{ 29 }(T)\) \(F_{ 31 }(T)\) \(F_{ 37 }(T)\) \(F_{ 41 }(T)\) \(F_{ 43 }(T)\) \(F_{ 47 }(T)\)
6-2200e3-1.1-c1e3-0-1 Modular form 2200.2.a.t $1$ $1 + 3 T + 6 T^{2} + 12 T^{3} + 18 T^{4} + 27 T^{5} + 27 T^{6}$ $1$ $1 + 3 T + 12 T^{2} + 46 T^{3} + 84 T^{4} + 147 T^{5} + 343 T^{6}$ $( 1 + T )^{3}$ $1 - 3 T + 18 T^{2} - 63 T^{3} + 234 T^{4} - 507 T^{5} + 2197 T^{6}$ $1 - 3 T + 6 T^{2} + 42 T^{3} + 102 T^{4} - 867 T^{5} + 4913 T^{6}$ $( 1 - T + 19 T^{2} )^{3}$ $1 + 6 T + 75 T^{2} + 271 T^{3} + 1725 T^{4} + 3174 T^{5} + 12167 T^{6}$ $1 - 12 T + 123 T^{2} - 727 T^{3} + 3567 T^{4} - 10092 T^{5} + 24389 T^{6}$ $1 + 3 T + 48 T^{2} + 259 T^{3} + 1488 T^{4} + 2883 T^{5} + 29791 T^{6}$ $1 + 12 T + 138 T^{2} + 884 T^{3} + 5106 T^{4} + 16428 T^{5} + 50653 T^{6}$ $1 - 6 T + 120 T^{2} - 472 T^{3} + 4920 T^{4} - 10086 T^{5} + 68921 T^{6}$ $1 - 3 T + 117 T^{2} - 246 T^{3} + 5031 T^{4} - 5547 T^{5} + 79507 T^{6}$ $1 + 12 T + 132 T^{2} + 1120 T^{3} + 6204 T^{4} + 26508 T^{5} + 103823 T^{6}$
6-4400e3-1.1-c1e3-0-1 Modular form 4400.2.a.ca $1$ $1 - 3 T + 6 T^{2} - 12 T^{3} + 18 T^{4} - 27 T^{5} + 27 T^{6}$ $1$ $1 - 3 T + 12 T^{2} - 46 T^{3} + 84 T^{4} - 147 T^{5} + 343 T^{6}$ $( 1 - T )^{3}$ $1 - 3 T + 18 T^{2} - 63 T^{3} + 234 T^{4} - 507 T^{5} + 2197 T^{6}$ $1 - 3 T + 6 T^{2} + 42 T^{3} + 102 T^{4} - 867 T^{5} + 4913 T^{6}$ $( 1 + T + 19 T^{2} )^{3}$ $1 - 6 T + 75 T^{2} - 271 T^{3} + 1725 T^{4} - 3174 T^{5} + 12167 T^{6}$ $1 - 12 T + 123 T^{2} - 727 T^{3} + 3567 T^{4} - 10092 T^{5} + 24389 T^{6}$ $1 - 3 T + 48 T^{2} - 259 T^{3} + 1488 T^{4} - 2883 T^{5} + 29791 T^{6}$ $1 + 12 T + 138 T^{2} + 884 T^{3} + 5106 T^{4} + 16428 T^{5} + 50653 T^{6}$ $1 - 6 T + 120 T^{2} - 472 T^{3} + 4920 T^{4} - 10086 T^{5} + 68921 T^{6}$ $1 + 3 T + 117 T^{2} + 246 T^{3} + 5031 T^{4} + 5547 T^{5} + 79507 T^{6}$ $1 - 12 T + 132 T^{2} - 1120 T^{3} + 6204 T^{4} - 26508 T^{5} + 103823 T^{6}$