LMFDB - L-function 40-384e20-1.1-c5e20-0-3

Dirichlet series

L(s) = 1

− 22·9^-s + 6.11e3·23^-s − 2.36e4·25^-s + 2.61e4·47^-s + 1.27e5·49^-s − 7.22e4·71^-s − 6.12e4·73^-s + 2.26e4·81^-s + 9.28e4·97^-s + 1.39e6·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 3.44e6·169^-s + 173^-s + 179^-s + 181^-s + 191^-s + 193^-s + 197^-s + 199^-s + ⋯

L(s) = 1

− 0.0905·9^-s + 2.40·23^-s − 7.56·25^-s + 1.72·47^-s + 7.58·49^-s − 1.70·71^-s − 1.34·73^-s + 0.383·81^-s + 1.00·97^-s + 8.68·121^-s + 5.50e−6·127^-s + 5.09e−6·131^-s + 4.55e−6·137^-s + 4.38e−6·139^-s + 3.69e−6·149^-s + 3.56e−6·151^-s + 3.23e−6·157^-s + 2.94e−6·163^-s + 2.77e−6·167^-s + 9.28·169^-s + 2.54e−6·173^-s + 2.33e−6·179^-s + 2.26e−6·181^-s + 1.98e−6·191^-s + 1.93e−6·193^-s + 1.83e−6·197^-s + 1.79e−6·199^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$40$
Conductor:	$2^{140} \cdot 3^{20}$
Sign:	$1$
Analytic conductor:	$6.16361\times 10^{35}$
Root analytic conductor:	$7.84776$
Motivic weight:	$5$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(40,\ 2^{140} \cdot 3^{20} ,\ ( \ : [5/2]^{20} ),\ 1 )$

Particular Values

$L(3)$	$\approx$	$18.18067038$
$L(\frac12)$	$\approx$	$18.18067038$
$L(\frac{7}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 + 22 T^{2} - 7385 p T^{4} - 81272 p^{2} T^{6} - 1029758 p^{6} T^{8} + 4685668 p^{10} T^{10} - 1029758 p^{16} T^{12} - 81272 p^{22} T^{14} - 7385 p^{31} T^{16} + 22 p^{40} T^{18} + p^{50} T^{20} $
good	5	$ ( 1 + 11818 T^{2} + 71488773 T^{4} + 301719010488 T^{6} + 1020428065064178 T^{8} + 3187442823228138492 T^{10} + 1020428065064178 p^{10} T^{12} + 301719010488 p^{20} T^{14} + 71488773 p^{30} T^{16} + 11818 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	7	$ ( 1 - 63774 T^{2} + 2508631325 T^{4} - 71829091413864 T^{6} + 1643928188403720658 T^{8} - $$30\!\cdots\!84$$ T^{10} + 1643928188403720658 p^{10} T^{12} - 71829091413864 p^{20} T^{14} + 2508631325 p^{30} T^{16} - 63774 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	11	$ ( 1 - 699642 T^{2} + 243730562789 T^{4} - 5167862023355496 p T^{6} + $$10\!\cdots\!46$$ T^{8} - $$17\!\cdots\!12$$ T^{10} + $$10\!\cdots\!46$$ p^{10} T^{12} - 5167862023355496 p^{21} T^{14} + 243730562789 p^{30} T^{16} - 699642 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	13	$ ( 1 - 1723346 T^{2} + 1796192221653 T^{4} - 1275220091760713112 T^{6} + $$69\!\cdots\!74$$ T^{8} - $$28\!\cdots\!24$$ T^{10} + $$69\!\cdots\!74$$ p^{10} T^{12} - 1275220091760713112 p^{20} T^{14} + 1796192221653 p^{30} T^{16} - 1723346 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	17	$ ( 1 - 5921674 T^{2} + 16210933829709 T^{4} - 28076383539701171064 T^{6} + $$36\!\cdots\!46$$ T^{8} - $$45\!\cdots\!60$$ T^{10} + $$36\!\cdots\!46$$ p^{10} T^{12} - 28076383539701171064 p^{20} T^{14} + 16210933829709 p^{30} T^{16} - 5921674 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	19	$ ( 1 + 11864886 T^{2} + 77718769794773 T^{4} + $$34\!\cdots\!92$$ T^{6} + $$11\!\cdots\!82$$ T^{8} + $$32\!\cdots\!32$$ T^{10} + $$11\!\cdots\!82$$ p^{10} T^{12} + $$34\!\cdots\!92$$ p^{20} T^{14} + 77718769794773 p^{30} T^{16} + 11864886 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	23	$ ( 1 - 1528 T + 16750163 T^{2} - 30494836000 T^{3} + 181299677525418 T^{4} - 239369149735345360 T^{5} + 181299677525418 p^{5} T^{6} - 30494836000 p^{10} T^{7} + 16750163 p^{15} T^{8} - 1528 p^{20} T^{9} + p^{25} T^{10} )^{4} $
	29	$ ( 1 + 108105178 T^{2} + 6217574540910645 T^{4} + $$24\!\cdots\!00$$ T^{6} + $$72\!\cdots\!38$$ T^{8} + $$16\!\cdots\!84$$ T^{10} + $$72\!\cdots\!38$$ p^{10} T^{12} + $$24\!\cdots\!00$$ p^{20} T^{14} + 6217574540910645 p^{30} T^{16} + 108105178 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	31	$ ( 1 - 124893838 T^{2} + 8535334798557357 T^{4} - $$40\!\cdots\!44$$ T^{6} + $$14\!\cdots\!34$$ T^{8} - $$45\!\cdots\!12$$ T^{10} + $$14\!\cdots\!34$$ p^{10} T^{12} - $$40\!\cdots\!44$$ p^{20} T^{14} + 8535334798557357 p^{30} T^{16} - 124893838 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	37	$ ( 1 - 483498626 T^{2} + 112886858733188229 T^{4} - $$16\!\cdots\!88$$ T^{6} + $$17\!\cdots\!54$$ T^{8} - $$14\!\cdots\!12$$ T^{10} + $$17\!\cdots\!54$$ p^{10} T^{12} - $$16\!\cdots\!88$$ p^{20} T^{14} + 112886858733188229 p^{30} T^{16} - 483498626 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	41	$ ( 1 - 683056346 T^{2} + 238166724811925181 T^{4} - $$55\!\cdots\!12$$ T^{6} + $$94\!\cdots\!30$$ T^{8} - $$12\!\cdots\!08$$ T^{10} + $$94\!\cdots\!30$$ p^{10} T^{12} - $$55\!\cdots\!12$$ p^{20} T^{14} + 238166724811925181 p^{30} T^{16} - 683056346 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	43	$ ( 1 + 8191410 p T^{2} + 125799691331078309 T^{4} + $$29\!\cdots\!88$$ T^{6} + $$57\!\cdots\!46$$ T^{8} + $$94\!\cdots\!72$$ T^{10} + $$57\!\cdots\!46$$ p^{10} T^{12} + $$29\!\cdots\!88$$ p^{20} T^{14} + 125799691331078309 p^{30} T^{16} + 8191410 p^{41} T^{18} + p^{50} T^{20} )^{2} $
	47	$ ( 1 - 6528 T + 767922859 T^{2} - 120901854720 p T^{3} + 300594027559513930 T^{4} - $$18\!\cdots\!36$$ T^{5} + 300594027559513930 p^{5} T^{6} - 120901854720 p^{11} T^{7} + 767922859 p^{15} T^{8} - 6528 p^{20} T^{9} + p^{25} T^{10} )^{4} $
	53	$ ( 1 + 1787609162 T^{2} + 1368832768041552357 T^{4} + $$73\!\cdots\!84$$ T^{6} + $$39\!\cdots\!02$$ T^{8} + $$19\!\cdots\!48$$ T^{10} + $$39\!\cdots\!02$$ p^{10} T^{12} + $$73\!\cdots\!84$$ p^{20} T^{14} + 1368832768041552357 p^{30} T^{16} + 1787609162 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	59	$ ( 1 - 5147616698 T^{2} + 12761348987329640709 T^{4} - $$20\!\cdots\!56$$ T^{6} + $$22\!\cdots\!06$$ T^{8} - $$18\!\cdots\!24$$ T^{10} + $$22\!\cdots\!06$$ p^{10} T^{12} - $$20\!\cdots\!56$$ p^{20} T^{14} + 12761348987329640709 p^{30} T^{16} - 5147616698 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	61	$ ( 1 - 1955556146 T^{2} + 1575682574427862965 T^{4} - $$12\!\cdots\!08$$ T^{6} + $$17\!\cdots\!70$$ T^{8} - $$19\!\cdots\!60$$ T^{10} + $$17\!\cdots\!70$$ p^{10} T^{12} - $$12\!\cdots\!08$$ p^{20} T^{14} + 1575682574427862965 p^{30} T^{16} - 1955556146 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	67	$ ( 1 + 7159517910 T^{2} + 342715510540507751 p T^{4} + $$43\!\cdots\!88$$ T^{6} + $$57\!\cdots\!22$$ T^{8} + $$72\!\cdots\!40$$ T^{10} + $$57\!\cdots\!22$$ p^{10} T^{12} + $$43\!\cdots\!88$$ p^{20} T^{14} + 342715510540507751 p^{31} T^{16} + 7159517910 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	71	$ ( 1 + 18056 T + 6774825731 T^{2} + 85449010534112 T^{3} + 21117998926544638698 T^{4} + $$20\!\cdots\!24$$ T^{5} + 21117998926544638698 p^{5} T^{6} + 85449010534112 p^{10} T^{7} + 6774825731 p^{15} T^{8} + 18056 p^{20} T^{9} + p^{25} T^{10} )^{4} $
	73	$ ( 1 + 15314 T + 4269004917 T^{2} + 139409155522200 T^{3} + 10772946279207978354 T^{4} + $$37\!\cdots\!88$$ T^{5} + 10772946279207978354 p^{5} T^{6} + 139409155522200 p^{10} T^{7} + 4269004917 p^{15} T^{8} + 15314 p^{20} T^{9} + p^{25} T^{10} )^{4} $
	79	$ ( 1 - 21221091630 T^{2} + $$21\!\cdots\!81$$ T^{4} - $$14\!\cdots\!56$$ T^{6} + $$68\!\cdots\!50$$ T^{8} - $$24\!\cdots\!92$$ T^{10} + $$68\!\cdots\!50$$ p^{10} T^{12} - $$14\!\cdots\!56$$ p^{20} T^{14} + $$21\!\cdots\!81$$ p^{30} T^{16} - 21221091630 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	83	$ ( 1 - 29570512106 T^{2} + $$41\!\cdots\!57$$ T^{4} - $$37\!\cdots\!56$$ T^{6} + $$23\!\cdots\!78$$ T^{8} - $$10\!\cdots\!16$$ T^{10} + $$23\!\cdots\!78$$ p^{10} T^{12} - $$37\!\cdots\!56$$ p^{20} T^{14} + $$41\!\cdots\!57$$ p^{30} T^{16} - 29570512106 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	89	$ ( 1 - 14639611546 T^{2} + $$12\!\cdots\!77$$ T^{4} - $$95\!\cdots\!20$$ T^{6} + $$59\!\cdots\!94$$ T^{8} - $$31\!\cdots\!32$$ T^{10} + $$59\!\cdots\!94$$ p^{10} T^{12} - $$95\!\cdots\!20$$ p^{20} T^{14} + $$12\!\cdots\!77$$ p^{30} T^{16} - 14639611546 p^{40} T^{18} + p^{50} T^{20} )^{2} $
	97	$ ( 1 - 23202 T + 14223120845 T^{2} + 159904888164072 T^{3} + $$18\!\cdots\!26$$ T^{4} + $$14\!\cdots\!00$$ T^{5} + $$18\!\cdots\!26$$ p^{5} T^{6} + 159904888164072 p^{10} T^{7} + 14223120845 p^{15} T^{8} - 23202 p^{20} T^{9} + p^{25} T^{10} )^{4} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−1.84687403718041399455846014228, −1.81903892565710898643682097232, −1.78230022555417223678423133839, −1.66415985665965804117540432036, −1.60041093538521950589345363598, −1.53885906212164590474221879121, −1.51499351829419733402638301527, −1.47424195506936382871063760277, −1.18507693355874128312060666372, −1.13164499588923528579915939216, −0.928015746372507847796980616538, −0.910580649343563328248446807242, −0.846579627700595925245763367581, −0.76475736366262834069398987739, −0.62710357309132161436960629217, −0.61347919064135652501018852058, −0.60913713957265843181553040461, −0.60263989787565815174328695222, −0.60068019769877229606578955246, −0.59131728471604477620162239775, −0.51221938169488612788646480889, −0.38278290375873712831285413058, −0.23726155572203822991481294890, −0.11838964934203209428105228704, −0.04774305056631611499042546569, 0.04774305056631611499042546569, 0.11838964934203209428105228704, 0.23726155572203822991481294890, 0.38278290375873712831285413058, 0.51221938169488612788646480889, 0.59131728471604477620162239775, 0.60068019769877229606578955246, 0.60263989787565815174328695222, 0.60913713957265843181553040461, 0.61347919064135652501018852058, 0.62710357309132161436960629217, 0.76475736366262834069398987739, 0.846579627700595925245763367581, 0.910580649343563328248446807242, 0.928015746372507847796980616538, 1.13164499588923528579915939216, 1.18507693355874128312060666372, 1.47424195506936382871063760277, 1.51499351829419733402638301527, 1.53885906212164590474221879121, 1.60041093538521950589345363598, 1.66415985665965804117540432036, 1.78230022555417223678423133839, 1.81903892565710898643682097232, 1.84687403718041399455846014228

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(3)\)	\(\approx\)	\(18.18067038\)
\(L(\frac12)\)	\(\approx\)	\(18.18067038\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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