LMFDB - L-function 28-325e14-1.1-c3e14-0-3

Dirichlet series

L(s) = 1

− 4·2^-s − 20·4^-s + 108·7^-s + 96·8^-s + 110·9^-s + 6·13^-s − 432·14^-s + 158·16^-s − 440·18^-s − 24·26^-s − 2.16e3·28^-s − 588·29^-s − 1.28e3·32^-s − 2.20e3·36^-s + 940·37^-s + 772·47^-s + 4.08e3·49^-s − 120·52^-s + 1.03e4·56^-s + 2.35e3·58^-s − 1.46e3·61^-s + 1.18e4·63^-s + 32·64^-s − 3.29e3·67^-s + 1.05e4·72^-s + 3.22e3·73^-s − 3.76e3·74^-s + ⋯

L(s) = 1

− 1.41·2^-s − 5/2·4^-s + 5.83·7^-s + 4.24·8^-s + 4.07·9^-s + 0.128·13^-s − 8.24·14^-s + 2.46·16^-s − 5.76·18^-s − 0.181·26^-s − 14.5·28^-s − 3.76·29^-s − 7.09·32^-s − 10.1·36^-s + 4.17·37^-s + 2.39·47^-s + 11.9·49^-s − 0.320·52^-s + 24.7·56^-s + 5.32·58^-s − 3.06·61^-s + 23.7·63^-s + 1/16·64^-s − 6.00·67^-s + 17.2·72^-s + 5.16·73^-s − 5.90·74^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{28} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{28} \cdot 13^{14}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$28$
Conductor:	$5^{28} \cdot 13^{14}$
Sign:	$1$
Analytic conductor:	$9.08849\times 10^{17}$
Root analytic conductor:	$4.37899$
Motivic weight:	$3$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(28,\ 5^{28} \cdot 13^{14} ,\ ( \ : [3/2]^{14} ),\ 1 )$

Particular Values

$L(2)$	$\approx$	$6.584672127$
$L(\frac12)$	$\approx$	$6.584672127$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	5	$ 1 $
	13	$ 1 - 6 T + 5495 T^{2} + 8420 p T^{3} + 1123301 p T^{4} + 5491462 p^{2} T^{5} + 10072419 p^{3} T^{6} + 102062680 p^{4} T^{7} + 10072419 p^{6} T^{8} + 5491462 p^{8} T^{9} + 1123301 p^{10} T^{10} + 8420 p^{13} T^{11} + 5495 p^{15} T^{12} - 6 p^{18} T^{13} + p^{21} T^{14} $
good	2	$ ( 1 + p T + p^{4} T^{2} + p^{5} T^{3} + 153 T^{4} + 55 p^{3} T^{5} + 207 p^{3} T^{6} + 153 p^{5} T^{7} + 207 p^{6} T^{8} + 55 p^{9} T^{9} + 153 p^{9} T^{10} + p^{17} T^{11} + p^{19} T^{12} + p^{19} T^{13} + p^{21} T^{14} )^{2} $
	3	$ 1 - 110 T^{2} + 871 p^{2} T^{4} - 422860 T^{6} + 18751589 T^{8} - 26302982 p^{3} T^{10} + 23341671931 T^{12} - 672312341672 T^{14} + 23341671931 p^{6} T^{16} - 26302982 p^{15} T^{18} + 18751589 p^{18} T^{20} - 422860 p^{24} T^{22} + 871 p^{32} T^{24} - 110 p^{36} T^{26} + p^{42} T^{28} $
	7	$ ( 1 - 54 T + 2333 T^{2} - 11476 p T^{3} + 2325761 T^{4} - 57742106 T^{5} + 1284512613 T^{6} - 25063331336 T^{7} + 1284512613 p^{3} T^{8} - 57742106 p^{6} T^{9} + 2325761 p^{9} T^{10} - 11476 p^{13} T^{11} + 2333 p^{15} T^{12} - 54 p^{18} T^{13} + p^{21} T^{14} )^{2} $
	11	$ 1 - 12358 T^{2} + 73222823 T^{4} - 279269174812 T^{6} + 776679460198925 T^{8} - 1686613805523637130 T^{10} + $$29\!\cdots\!11$$ T^{12} - $$43\!\cdots\!40$$ T^{14} + $$29\!\cdots\!11$$ p^{6} T^{16} - 1686613805523637130 p^{12} T^{18} + 776679460198925 p^{18} T^{20} - 279269174812 p^{24} T^{22} + 73222823 p^{30} T^{24} - 12358 p^{36} T^{26} + p^{42} T^{28} $
	17	$ 1 - 36730 T^{2} + 41296795 p T^{4} - 9171354963012 T^{6} + 90720639059398969 T^{8} - $$71\!\cdots\!46$$ T^{10} + $$27\!\cdots\!23$$ p T^{12} - $$24\!\cdots\!24$$ T^{14} + $$27\!\cdots\!23$$ p^{7} T^{16} - $$71\!\cdots\!46$$ p^{12} T^{18} + 90720639059398969 p^{18} T^{20} - 9171354963012 p^{24} T^{22} + 41296795 p^{31} T^{24} - 36730 p^{36} T^{26} + p^{42} T^{28} $
	19	$ 1 - 56638 T^{2} + 1587803255 T^{4} - 29442000078460 T^{6} + 406369274698402317 T^{8} - $$44\!\cdots\!38$$ T^{10} + $$39\!\cdots\!83$$ T^{12} - $$29\!\cdots\!00$$ T^{14} + $$39\!\cdots\!83$$ p^{6} T^{16} - $$44\!\cdots\!38$$ p^{12} T^{18} + 406369274698402317 p^{18} T^{20} - 29442000078460 p^{24} T^{22} + 1587803255 p^{30} T^{24} - 56638 p^{36} T^{26} + p^{42} T^{28} $
	23	$ 1 - 78514 T^{2} + 3204753527 T^{4} - 92013049435668 T^{6} + 2062719255318115621 T^{8} - $$37\!\cdots\!62$$ T^{10} + $$58\!\cdots\!55$$ T^{12} - $$76\!\cdots\!12$$ T^{14} + $$58\!\cdots\!55$$ p^{6} T^{16} - $$37\!\cdots\!62$$ p^{12} T^{18} + 2062719255318115621 p^{18} T^{20} - 92013049435668 p^{24} T^{22} + 3204753527 p^{30} T^{24} - 78514 p^{36} T^{26} + p^{42} T^{28} $
	29	$ ( 1 + 294 T + 109035 T^{2} + 20517020 T^{3} + 5002413109 T^{4} + 773140090362 T^{5} + 158882226378071 T^{6} + 21393846073167368 T^{7} + 158882226378071 p^{3} T^{8} + 773140090362 p^{6} T^{9} + 5002413109 p^{9} T^{10} + 20517020 p^{12} T^{11} + 109035 p^{15} T^{12} + 294 p^{18} T^{13} + p^{21} T^{14} )^{2} $
	31	$ 1 - 250346 T^{2} + 31735294927 T^{4} - 2684453200058676 T^{6} + $$16\!\cdots\!97$$ T^{8} - $$83\!\cdots\!22$$ T^{10} + $$33\!\cdots\!03$$ T^{12} - $$10\!\cdots\!28$$ T^{14} + $$33\!\cdots\!03$$ p^{6} T^{16} - $$83\!\cdots\!22$$ p^{12} T^{18} + $$16\!\cdots\!97$$ p^{18} T^{20} - 2684453200058676 p^{24} T^{22} + 31735294927 p^{30} T^{24} - 250346 p^{36} T^{26} + p^{42} T^{28} $
	37	$ ( 1 - 470 T + 318327 T^{2} - 102025324 T^{3} + 42061831113 T^{4} - 10683186485418 T^{5} + 3318516181175575 T^{6} - 681773941683913832 T^{7} + 3318516181175575 p^{3} T^{8} - 10683186485418 p^{6} T^{9} + 42061831113 p^{9} T^{10} - 102025324 p^{12} T^{11} + 318327 p^{15} T^{12} - 470 p^{18} T^{13} + p^{21} T^{14} )^{2} $
	41	$ 1 - 271414 T^{2} + 44449267067 T^{4} - 5606468972235996 T^{6} + $$60\!\cdots\!93$$ T^{8} - $$55\!\cdots\!90$$ T^{10} + $$45\!\cdots\!39$$ T^{12} - $$33\!\cdots\!80$$ T^{14} + $$45\!\cdots\!39$$ p^{6} T^{16} - $$55\!\cdots\!90$$ p^{12} T^{18} + $$60\!\cdots\!93$$ p^{18} T^{20} - 5606468972235996 p^{24} T^{22} + 44449267067 p^{30} T^{24} - 271414 p^{36} T^{26} + p^{42} T^{28} $
	43	$ 1 - 687302 T^{2} + 241180616687 T^{4} - 56543572112880940 T^{6} + $$98\!\cdots\!81$$ T^{8} - $$13\!\cdots\!22$$ T^{10} + $$14\!\cdots\!15$$ T^{12} - $$12\!\cdots\!72$$ T^{14} + $$14\!\cdots\!15$$ p^{6} T^{16} - $$13\!\cdots\!22$$ p^{12} T^{18} + $$98\!\cdots\!81$$ p^{18} T^{20} - 56543572112880940 p^{24} T^{22} + 241180616687 p^{30} T^{24} - 687302 p^{36} T^{26} + p^{42} T^{28} $
	47	$ ( 1 - 386 T + 471117 T^{2} - 120513092 T^{3} + 99001425209 T^{4} - 19759245910542 T^{5} + 14080854648355133 T^{6} - 2390748635870883736 T^{7} + 14080854648355133 p^{3} T^{8} - 19759245910542 p^{6} T^{9} + 99001425209 p^{9} T^{10} - 120513092 p^{12} T^{11} + 471117 p^{15} T^{12} - 386 p^{18} T^{13} + p^{21} T^{14} )^{2} $
	53	$ 1 - 996334 T^{2} + 484318931747 T^{4} - 155483656952480428 T^{6} + $$37\!\cdots\!17$$ T^{8} - $$72\!\cdots\!94$$ T^{10} + $$12\!\cdots\!27$$ T^{12} - $$18\!\cdots\!08$$ T^{14} + $$12\!\cdots\!27$$ p^{6} T^{16} - $$72\!\cdots\!94$$ p^{12} T^{18} + $$37\!\cdots\!17$$ p^{18} T^{20} - 155483656952480428 p^{24} T^{22} + 484318931747 p^{30} T^{24} - 996334 p^{36} T^{26} + p^{42} T^{28} $
	59	$ 1 - 1657502 T^{2} + 1415661698375 T^{4} - 810149316940284796 T^{6} + $$34\!\cdots\!49$$ T^{8} - $$11\!\cdots\!54$$ T^{10} + $$31\!\cdots\!19$$ T^{12} - $$71\!\cdots\!44$$ T^{14} + $$31\!\cdots\!19$$ p^{6} T^{16} - $$11\!\cdots\!54$$ p^{12} T^{18} + $$34\!\cdots\!49$$ p^{18} T^{20} - 810149316940284796 p^{24} T^{22} + 1415661698375 p^{30} T^{24} - 1657502 p^{36} T^{26} + p^{42} T^{28} $
	61	$ ( 1 + 730 T + 1242147 T^{2} + 831341412 T^{3} + 708901440253 T^{4} + 424396208710438 T^{5} + 244692387212101815 T^{6} + $$12\!\cdots\!08$$ T^{7} + 244692387212101815 p^{3} T^{8} + 424396208710438 p^{6} T^{9} + 708901440253 p^{9} T^{10} + 831341412 p^{12} T^{11} + 1242147 p^{15} T^{12} + 730 p^{18} T^{13} + p^{21} T^{14} )^{2} $
	67	$ ( 1 + 1646 T + 2002545 T^{2} + 1573617780 T^{3} + 1066920817153 T^{4} + 538266837503922 T^{5} + 275569421024656297 T^{6} + $$13\!\cdots\!40$$ T^{7} + 275569421024656297 p^{3} T^{8} + 538266837503922 p^{6} T^{9} + 1066920817153 p^{9} T^{10} + 1573617780 p^{12} T^{11} + 2002545 p^{15} T^{12} + 1646 p^{18} T^{13} + p^{21} T^{14} )^{2} $
	71	$ 1 - 2287442 T^{2} + 2847434457311 T^{4} - 2474880637852592724 T^{6} + $$16\!\cdots\!81$$ T^{8} - $$91\!\cdots\!26$$ T^{10} + $$42\!\cdots\!87$$ T^{12} - $$16\!\cdots\!16$$ T^{14} + $$42\!\cdots\!87$$ p^{6} T^{16} - $$91\!\cdots\!26$$ p^{12} T^{18} + $$16\!\cdots\!81$$ p^{18} T^{20} - 2474880637852592724 p^{24} T^{22} + 2847434457311 p^{30} T^{24} - 2287442 p^{36} T^{26} + p^{42} T^{28} $
	73	$ ( 1 - 1610 T + 2792139 T^{2} - 37934564 p T^{3} + 2944910965601 T^{4} - 2243019397603590 T^{5} + 1811447209789220275 T^{6} - $$11\!\cdots\!16$$ T^{7} + 1811447209789220275 p^{3} T^{8} - 2243019397603590 p^{6} T^{9} + 2944910965601 p^{9} T^{10} - 37934564 p^{13} T^{11} + 2792139 p^{15} T^{12} - 1610 p^{18} T^{13} + p^{21} T^{14} )^{2} $
	79	$ ( 1 - 512 T + 2045465 T^{2} - 1210402496 T^{3} + 2161877892581 T^{4} - 1326231242967808 T^{5} + 1498346935045947885 T^{6} - $$83\!\cdots\!48$$ T^{7} + 1498346935045947885 p^{3} T^{8} - 1326231242967808 p^{6} T^{9} + 2161877892581 p^{9} T^{10} - 1210402496 p^{12} T^{11} + 2045465 p^{15} T^{12} - 512 p^{18} T^{13} + p^{21} T^{14} )^{2} $
	83	$ ( 1 + 226 T + 551697 T^{2} - 511708068 T^{3} + 326757867457 T^{4} - 337380082800738 T^{5} + 214284091113451049 T^{6} - $$30\!\cdots\!84$$ T^{7} + 214284091113451049 p^{3} T^{8} - 337380082800738 p^{6} T^{9} + 326757867457 p^{9} T^{10} - 511708068 p^{12} T^{11} + 551697 p^{15} T^{12} + 226 p^{18} T^{13} + p^{21} T^{14} )^{2} $
	89	$ 1 - 5109934 T^{2} + 12320790316939 T^{4} - 19197704418883074284 T^{6} + $$22\!\cdots\!25$$ T^{8} - $$21\!\cdots\!90$$ T^{10} + $$17\!\cdots\!59$$ T^{12} - $$13\!\cdots\!92$$ T^{14} + $$17\!\cdots\!59$$ p^{6} T^{16} - $$21\!\cdots\!90$$ p^{12} T^{18} + $$22\!\cdots\!25$$ p^{18} T^{20} - 19197704418883074284 p^{24} T^{22} + 12320790316939 p^{30} T^{24} - 5109934 p^{36} T^{26} + p^{42} T^{28} $
	97	$ ( 1 + 1982 T + 4421403 T^{2} + 5685747788 T^{3} + 7421769693833 T^{4} + 7694710573386402 T^{5} + 7739121271091709979 T^{6} + $$74\!\cdots\!04$$ T^{7} + 7739121271091709979 p^{3} T^{8} + 7694710573386402 p^{6} T^{9} + 7421769693833 p^{9} T^{10} + 5685747788 p^{12} T^{11} + 4421403 p^{15} T^{12} + 1982 p^{18} T^{13} + p^{21} T^{14} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−3.05557567277538039150018633288, −2.87813965575833789482074598897, −2.81836832896434153655907618913, −2.48575670172825518322646321815, −2.43199806787242713951062471143, −2.38385779074776564244516254017, −2.37709765489022617010114747866, −2.11355560584711313574437446677, −1.99910674953034886523875519861, −1.86164024032385107667169171893, −1.84769537987803970267498628122, −1.65270965565933347725937650920, −1.59242131216077303440746875421, −1.43023872237056155209701713505, −1.37968352521761517422870580483, −1.36357311151824406247620965545, −1.35166667452475675259149872875, −1.20138469624739790102201642636, −1.14178370458727739524561071665, −0.75485113101469053759113137251, −0.60287573092453937449055728504, −0.55947467894235674865704986802, −0.33865616466534539526183719149, −0.32972245259846679840215035406, −0.15175086396277666241705094482, 0.15175086396277666241705094482, 0.32972245259846679840215035406, 0.33865616466534539526183719149, 0.55947467894235674865704986802, 0.60287573092453937449055728504, 0.75485113101469053759113137251, 1.14178370458727739524561071665, 1.20138469624739790102201642636, 1.35166667452475675259149872875, 1.36357311151824406247620965545, 1.37968352521761517422870580483, 1.43023872237056155209701713505, 1.59242131216077303440746875421, 1.65270965565933347725937650920, 1.84769537987803970267498628122, 1.86164024032385107667169171893, 1.99910674953034886523875519861, 2.11355560584711313574437446677, 2.37709765489022617010114747866, 2.38385779074776564244516254017, 2.43199806787242713951062471143, 2.48575670172825518322646321815, 2.81836832896434153655907618913, 2.87813965575833789482074598897, 3.05557567277538039150018633288

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 $\F_{q}$
Belyi maps

\(L(2)\)	\(\approx\)	\(6.584672127\)
\(L(\frac12)\)	\(\approx\)	\(6.584672127\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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