LMFDB - L-function Euler product search

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Lowest zero	Conductor	Analytic conductor	Central character	Analytic rank

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

Results (23 matches)

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		$L(s) = \prod_p F_p(p^{-s})^{-1}$
Label	Origin	$F_{ 2 }(T)$	$F_{ 3 }(T)$	$F_{ 5 }(T)$	$F_{ 7 }(T)$
6-267e3-1.1-c1e3-0-1	Modular form 267.2.a.d	$1 + 3 T^{2} + T^{3} + 6 T^{4} + 8 T^{6}$	$( 1 + T )^{3}$	$1 + 3 T + 9 T^{2} + 31 T^{3} + 45 T^{4} + 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-801e3-1.1-c1e3-0-2	Modular form 801.2.a.f	$1 + 3 T^{2} - T^{3} + 6 T^{4} + 8 T^{6}$	$1$	$1 - 3 T + 9 T^{2} - 31 T^{3} + 45 T^{4} - 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-37e6-1.1-c1e3-0-0	Modular form 1369.2.a.k	$1 + 3 T^{2} + T^{3} + 6 T^{4} + 8 T^{6}$	$1 + 3 T + 9 T^{2} + 15 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$1 + 3 T^{2} - 8 T^{3} + 15 T^{4} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-37e6-1.1-c1e3-0-1	Modular form 1369.2.a.j	$1 + 3 T^{2} - T^{3} + 6 T^{4} + 8 T^{6}$	$1 + 3 T + 9 T^{2} + 15 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$1 + 3 T^{2} + 8 T^{3} + 15 T^{4} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-1458e3-1.1-c1e3-0-1	Modular form 1458.2.a.a	$( 1 + T )^{3}$	$1$	$1 - 3 T + 15 T^{2} - 27 T^{3} + 75 T^{4} - 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-1458e3-1.1-c1e3-0-2	Modular form 1458.2.a.d	$( 1 - T )^{3}$	$1$	$1 + 3 T + 15 T^{2} + 27 T^{3} + 75 T^{4} + 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-2470e3-1.1-c1e3-0-5	Modular form 2470.2.a.n	$( 1 - T )^{3}$	$1 + 3 T + 9 T^{2} + 17 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$( 1 - T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-2658e3-1.1-c1e3-0-1	Modular form 2658.2.a.i	$( 1 - T )^{3}$	$( 1 - T )^{3}$	$1 + 3 T + 15 T^{2} + 29 T^{3} + 75 T^{4} + 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-3179e3-1.1-c1e3-0-3	Modular form 3179.2.a.q	$1 + 3 T + 6 T^{2} + 9 T^{3} + 12 T^{4} + 12 T^{5} + 8 T^{6}$	$1 + 6 T^{2} - T^{3} + 18 T^{4} + 27 T^{6}$	$1 + 3 T + 15 T^{2} + 27 T^{3} + 75 T^{4} + 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-3971e3-1.1-c1e3-0-0	Modular form 3971.2.a.e	$1 + 3 T + 6 T^{2} + 9 T^{3} + 12 T^{4} + 12 T^{5} + 8 T^{6}$	$( 1 + 2 T + 3 T^{2} )^{3}$	$1 + 6 T + 24 T^{2} + 63 T^{3} + 120 T^{4} + 150 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-3971e3-1.1-c1e3-0-1	Modular form 3971.2.a.f	$1 - 3 T + 6 T^{2} - 9 T^{3} + 12 T^{4} - 12 T^{5} + 8 T^{6}$	$( 1 - 2 T + 3 T^{2} )^{3}$	$1 + 6 T + 24 T^{2} + 63 T^{3} + 120 T^{4} + 150 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-4335e3-1.1-c1e3-0-6	Modular form 4335.2.a.t	$1 + 3 T^{2} - T^{3} + 6 T^{4} + 8 T^{6}$	$( 1 - T )^{3}$	$( 1 - T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-4908e3-1.1-c1e3-0-0	Modular form 4908.2.a.c	$1$	$( 1 - T )^{3}$	$1 + 6 T + 24 T^{2} + 61 T^{3} + 120 T^{4} + 150 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-5618e3-1.1-c1e3-0-3	Modular form 5618.2.a.p	$( 1 - T )^{3}$	$1 + 3 T + 9 T^{2} + 17 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$1 + 3 T + 15 T^{2} + 27 T^{3} + 75 T^{4} + 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-5618e3-1.1-c1e3-0-5	Modular form 5618.2.a.o	$( 1 + T )^{3}$	$1 - 3 T + 9 T^{2} - 17 T^{3} + 27 T^{4} - 27 T^{5} + 27 T^{6}$	$1 - 3 T + 15 T^{2} - 27 T^{3} + 75 T^{4} - 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-5830e3-1.1-c1e3-0-1	Modular form 5830.2.a.q	$( 1 + T )^{3}$	$1 + 3 T + 9 T^{2} + 17 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$( 1 - T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-6532e3-1.1-c1e3-0-0	Modular form 6532.2.a.a	$1$	$1 + 3 T + 9 T^{2} + 15 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$( 1 + 2 T + 5 T^{2} )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-7290e3-1.1-c1e3-0-1	Modular form 7290.2.a.a	$( 1 + T )^{3}$	$1$	$( 1 - T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-7290e3-1.1-c1e3-0-5	Modular form 7290.2.a.d	$( 1 - T )^{3}$	$1$	$( 1 + T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-7974e3-1.1-c1e3-0-0	Modular form 7974.2.a.r	$( 1 + T )^{3}$	$1$	$1 - 3 T + 15 T^{2} - 29 T^{3} + 75 T^{4} - 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-8670e3-1.1-c1e3-0-7	Modular form 8670.2.a.bs	$( 1 - T )^{3}$	$( 1 - T )^{3}$	$( 1 - T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-9477e3-1.1-c1e3-0-0	Modular form 9477.2.a.b	$1 + 3 T^{2} - T^{3} + 6 T^{4} + 8 T^{6}$	$1$	$1 + 3 T^{2} - 8 T^{3} + 15 T^{4} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-9477e3-1.1-c1e3-0-1	Modular form 9477.2.a.a	$1 + 3 T^{2} + T^{3} + 6 T^{4} + 8 T^{6}$	$1$	$1 + 3 T^{2} + 8 T^{3} + 15 T^{4} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$

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Results (23 matches)

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

		$L(s) = \prod_p F_p(p^{-s})^{-1}$
Label	Origin	\(F_{ 2 }(T)\)	\(F_{ 3 }(T)\)	\(F_{ 5 }(T)\)	\(F_{ 7 }(T)\)
6-267e3-1.1-c1e3-0-1	Modular form 267.2.a.d	$1 + 3 T^{2} + T^{3} + 6 T^{4} + 8 T^{6}$	$( 1 + T )^{3}$	$1 + 3 T + 9 T^{2} + 31 T^{3} + 45 T^{4} + 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-801e3-1.1-c1e3-0-2	Modular form 801.2.a.f	$1 + 3 T^{2} - T^{3} + 6 T^{4} + 8 T^{6}$	$1$	$1 - 3 T + 9 T^{2} - 31 T^{3} + 45 T^{4} - 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-37e6-1.1-c1e3-0-0	Modular form 1369.2.a.k	$1 + 3 T^{2} + T^{3} + 6 T^{4} + 8 T^{6}$	$1 + 3 T + 9 T^{2} + 15 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$1 + 3 T^{2} - 8 T^{3} + 15 T^{4} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-37e6-1.1-c1e3-0-1	Modular form 1369.2.a.j	$1 + 3 T^{2} - T^{3} + 6 T^{4} + 8 T^{6}$	$1 + 3 T + 9 T^{2} + 15 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$1 + 3 T^{2} + 8 T^{3} + 15 T^{4} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-1458e3-1.1-c1e3-0-1	Modular form 1458.2.a.a	$( 1 + T )^{3}$	$1$	$1 - 3 T + 15 T^{2} - 27 T^{3} + 75 T^{4} - 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-1458e3-1.1-c1e3-0-2	Modular form 1458.2.a.d	$( 1 - T )^{3}$	$1$	$1 + 3 T + 15 T^{2} + 27 T^{3} + 75 T^{4} + 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-2470e3-1.1-c1e3-0-5	Modular form 2470.2.a.n	$( 1 - T )^{3}$	$1 + 3 T + 9 T^{2} + 17 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$( 1 - T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-2658e3-1.1-c1e3-0-1	Modular form 2658.2.a.i	$( 1 - T )^{3}$	$( 1 - T )^{3}$	$1 + 3 T + 15 T^{2} + 29 T^{3} + 75 T^{4} + 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-3179e3-1.1-c1e3-0-3	Modular form 3179.2.a.q	$1 + 3 T + 6 T^{2} + 9 T^{3} + 12 T^{4} + 12 T^{5} + 8 T^{6}$	$1 + 6 T^{2} - T^{3} + 18 T^{4} + 27 T^{6}$	$1 + 3 T + 15 T^{2} + 27 T^{3} + 75 T^{4} + 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-3971e3-1.1-c1e3-0-0	Modular form 3971.2.a.e	$1 + 3 T + 6 T^{2} + 9 T^{3} + 12 T^{4} + 12 T^{5} + 8 T^{6}$	$( 1 + 2 T + 3 T^{2} )^{3}$	$1 + 6 T + 24 T^{2} + 63 T^{3} + 120 T^{4} + 150 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-3971e3-1.1-c1e3-0-1	Modular form 3971.2.a.f	$1 - 3 T + 6 T^{2} - 9 T^{3} + 12 T^{4} - 12 T^{5} + 8 T^{6}$	$( 1 - 2 T + 3 T^{2} )^{3}$	$1 + 6 T + 24 T^{2} + 63 T^{3} + 120 T^{4} + 150 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-4335e3-1.1-c1e3-0-6	Modular form 4335.2.a.t	$1 + 3 T^{2} - T^{3} + 6 T^{4} + 8 T^{6}$	$( 1 - T )^{3}$	$( 1 - T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-4908e3-1.1-c1e3-0-0	Modular form 4908.2.a.c	$1$	$( 1 - T )^{3}$	$1 + 6 T + 24 T^{2} + 61 T^{3} + 120 T^{4} + 150 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-5618e3-1.1-c1e3-0-3	Modular form 5618.2.a.p	$( 1 - T )^{3}$	$1 + 3 T + 9 T^{2} + 17 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$1 + 3 T + 15 T^{2} + 27 T^{3} + 75 T^{4} + 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-5618e3-1.1-c1e3-0-5	Modular form 5618.2.a.o	$( 1 + T )^{3}$	$1 - 3 T + 9 T^{2} - 17 T^{3} + 27 T^{4} - 27 T^{5} + 27 T^{6}$	$1 - 3 T + 15 T^{2} - 27 T^{3} + 75 T^{4} - 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-5830e3-1.1-c1e3-0-1	Modular form 5830.2.a.q	$( 1 + T )^{3}$	$1 + 3 T + 9 T^{2} + 17 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$( 1 - T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-6532e3-1.1-c1e3-0-0	Modular form 6532.2.a.a	$1$	$1 + 3 T + 9 T^{2} + 15 T^{3} + 27 T^{4} + 27 T^{5} + 27 T^{6}$	$( 1 + 2 T + 5 T^{2} )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-7290e3-1.1-c1e3-0-1	Modular form 7290.2.a.a	$( 1 + T )^{3}$	$1$	$( 1 - T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-7290e3-1.1-c1e3-0-5	Modular form 7290.2.a.d	$( 1 - T )^{3}$	$1$	$( 1 + T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-7974e3-1.1-c1e3-0-0	Modular form 7974.2.a.r	$( 1 + T )^{3}$	$1$	$1 - 3 T + 15 T^{2} - 29 T^{3} + 75 T^{4} - 75 T^{5} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-8670e3-1.1-c1e3-0-7	Modular form 8670.2.a.bs	$( 1 - T )^{3}$	$( 1 - T )^{3}$	$( 1 - T )^{3}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-9477e3-1.1-c1e3-0-0	Modular form 9477.2.a.b	$1 + 3 T^{2} - T^{3} + 6 T^{4} + 8 T^{6}$	$1$	$1 + 3 T^{2} - 8 T^{3} + 15 T^{4} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$
6-9477e3-1.1-c1e3-0-1	Modular form 9477.2.a.a	$1 + 3 T^{2} + T^{3} + 6 T^{4} + 8 T^{6}$	$1$	$1 + 3 T^{2} + 8 T^{3} + 15 T^{4} + 125 T^{6}$	$1 + 6 T + 30 T^{2} + 85 T^{3} + 210 T^{4} + 294 T^{5} + 343 T^{6}$