LMFDB - L-function 32-42e32-1.1-c3e16-0-2

Dirichlet series

L(s) = 1

+ 72·19^-s + 394·25^-s + 708·31^-s + 76·37^-s + 1.40e3·43^-s + 1.63e3·61^-s − 1.52e3·67^-s + 2.70e3·73^-s − 364·79^-s + 4.99e3·103^-s + 772·109^-s − 5.84e3·121^-s + 127^-s + 131^-s + 137^-s + 139^-s + 149^-s + 151^-s + 157^-s + 163^-s + 167^-s + 2.95e4·169^-s + 173^-s + 179^-s + 181^-s + 191^-s + 193^-s + ⋯

L(s) = 1

+ 0.869·19^-s + 3.15·25^-s + 4.10·31^-s + 0.337·37^-s + 4.99·43^-s + 3.42·61^-s − 2.78·67^-s + 4.32·73^-s − 0.518·79^-s + 4.77·103^-s + 0.678·109^-s − 4.38·121^-s + 0.000698·127^-s + 0.000666·131^-s + 0.000623·137^-s + 0.000610·139^-s + 0.000549·149^-s + 0.000538·151^-s + 0.000508·157^-s + 0.000480·163^-s + 0.000463·167^-s + 13.4·169^-s + 0.000439·173^-s + 0.000417·179^-s + 0.000410·181^-s + 0.000378·191^-s + 0.000372·193^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$32$
Conductor:	$2^{32} \cdot 3^{32} \cdot 7^{32}$
Sign:	$1$
Analytic conductor:	$1.89598\times 10^{32}$
Root analytic conductor:	$10.2019$
Motivic weight:	$3$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(32,\ 2^{32} \cdot 3^{32} \cdot 7^{32} ,\ ( \ : [3/2]^{16} ),\ 1 )$

Particular Values

$L(2)$	$\approx$	$1917.067563$
$L(\frac12)$	$\approx$	$1917.067563$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	7	$ 1 $
good	5	$ 1 - 394 T^{2} + 79383 T^{4} - 13135862 T^{6} + 1573353173 T^{8} - 72279808932 T^{10} - 11170130994038 T^{12} + 735694168015888 p T^{14} - 600595446685632534 T^{16} + 735694168015888 p^{7} T^{18} - 11170130994038 p^{12} T^{20} - 72279808932 p^{18} T^{22} + 1573353173 p^{24} T^{24} - 13135862 p^{30} T^{26} + 79383 p^{36} T^{28} - 394 p^{42} T^{30} + p^{48} T^{32} $
	11	$ 1 + 5842 T^{2} + 17711847 T^{4} + 34861793198 T^{6} + 46760130210965 T^{8} + 35906625944396532 T^{10} - 10800538135038597014 T^{12} - $$85\!\cdots\!92$$ T^{14} - $$15\!\cdots\!02$$ T^{16} - $$85\!\cdots\!92$$ p^{6} T^{18} - 10800538135038597014 p^{12} T^{20} + 35906625944396532 p^{18} T^{22} + 46760130210965 p^{24} T^{24} + 34861793198 p^{30} T^{26} + 17711847 p^{36} T^{28} + 5842 p^{42} T^{30} + p^{48} T^{32} $
	13	$ ( 1 - 14750 T^{2} + 100165933 T^{4} - 31441942490 p T^{6} + 1095322120237240 T^{8} - 31441942490 p^{7} T^{10} + 100165933 p^{12} T^{12} - 14750 p^{18} T^{14} + p^{24} T^{16} )^{2} $
	17	$ 1 - 17908 T^{2} + 183721800 T^{4} - 1337837599208 T^{6} + 6758666240315906 T^{8} - 17525131900492504332 T^{10} - $$47\!\cdots\!28$$ T^{12} + $$88\!\cdots\!12$$ T^{14} - $$56\!\cdots\!25$$ T^{16} + $$88\!\cdots\!12$$ p^{6} T^{18} - $$47\!\cdots\!28$$ p^{12} T^{20} - 17525131900492504332 p^{18} T^{22} + 6758666240315906 p^{24} T^{24} - 1337837599208 p^{30} T^{26} + 183721800 p^{36} T^{28} - 17908 p^{42} T^{30} + p^{48} T^{32} $
	19	$ ( 1 - 36 T + 12655 T^{2} - 440028 T^{3} + 69273163 T^{4} - 128221056 T^{5} - 8891738156 p T^{6} + 30396011988456 T^{7} - 3052907831237222 T^{8} + 30396011988456 p^{3} T^{9} - 8891738156 p^{7} T^{10} - 128221056 p^{9} T^{11} + 69273163 p^{12} T^{12} - 440028 p^{15} T^{13} + 12655 p^{18} T^{14} - 36 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	23	$ 1 + 25876 T^{2} + 243513432 T^{4} + 1681286205992 T^{6} + 1505945652861218 T^{8} - $$37\!\cdots\!76$$ T^{10} - $$25\!\cdots\!88$$ T^{12} + $$23\!\cdots\!32$$ p^{2} T^{14} + $$15\!\cdots\!91$$ T^{16} + $$23\!\cdots\!32$$ p^{8} T^{18} - $$25\!\cdots\!88$$ p^{12} T^{20} - $$37\!\cdots\!76$$ p^{18} T^{22} + 1505945652861218 p^{24} T^{24} + 1681286205992 p^{30} T^{26} + 243513432 p^{36} T^{28} + 25876 p^{42} T^{30} + p^{48} T^{32} $
	29	$ ( 1 - 2138 p T^{2} + 2900130409 T^{4} - 97031213045314 T^{6} + 2790721647797586244 T^{8} - 97031213045314 p^{6} T^{10} + 2900130409 p^{12} T^{12} - 2138 p^{19} T^{14} + p^{24} T^{16} )^{2} $
	31	$ ( 1 - 354 T + 75094 T^{2} - 11795988 T^{3} + 883174573 T^{4} - 3505223700 T^{5} - 50834565509222 T^{6} + 19486802838799662 T^{7} - 3853637391311667068 T^{8} + 19486802838799662 p^{3} T^{9} - 50834565509222 p^{6} T^{10} - 3505223700 p^{9} T^{11} + 883174573 p^{12} T^{12} - 11795988 p^{15} T^{13} + 75094 p^{18} T^{14} - 354 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	37	$ ( 1 - 38 T - 91851 T^{2} - 20169358 T^{3} + 5124013139 T^{4} + 1492587433200 T^{5} + 200369748531112 T^{6} - 60212649379130144 T^{7} - 16861961095425650754 T^{8} - 60212649379130144 p^{3} T^{9} + 200369748531112 p^{6} T^{10} + 1492587433200 p^{9} T^{11} + 5124013139 p^{12} T^{12} - 20169358 p^{15} T^{13} - 91851 p^{18} T^{14} - 38 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	41	$ ( 1 + 311308 T^{2} + 49778800408 T^{4} + 5249339160024580 T^{6} + $$41\!\cdots\!42$$ T^{8} + 5249339160024580 p^{6} T^{10} + 49778800408 p^{12} T^{12} + 311308 p^{18} T^{14} + p^{24} T^{16} )^{2} $
	43	$ ( 1 - 352 T + 171577 T^{2} - 15193816 T^{3} + 8385609238 T^{4} - 15193816 p^{3} T^{5} + 171577 p^{6} T^{6} - 352 p^{9} T^{7} + p^{12} T^{8} )^{4} $
	47	$ 1 - 307672 T^{2} + 35469631884 T^{4} - 1804225864402640 T^{6} + 27431766123470856938 T^{8} + $$11\!\cdots\!68$$ T^{10} - $$25\!\cdots\!40$$ T^{12} + $$15\!\cdots\!32$$ T^{14} - $$22\!\cdots\!25$$ T^{16} + $$15\!\cdots\!32$$ p^{6} T^{18} - $$25\!\cdots\!40$$ p^{12} T^{20} + $$11\!\cdots\!68$$ p^{18} T^{22} + 27431766123470856938 p^{24} T^{24} - 1804225864402640 p^{30} T^{26} + 35469631884 p^{36} T^{28} - 307672 p^{42} T^{30} + p^{48} T^{32} $
	53	$ 1 + 938962 T^{2} + 467576763291 T^{4} + 165402918375811934 T^{6} + $$46\!\cdots\!33$$ T^{8} + $$10\!\cdots\!96$$ T^{10} + $$22\!\cdots\!82$$ T^{12} + $$39\!\cdots\!84$$ T^{14} + $$62\!\cdots\!74$$ T^{16} + $$39\!\cdots\!84$$ p^{6} T^{18} + $$22\!\cdots\!82$$ p^{12} T^{20} + $$10\!\cdots\!96$$ p^{18} T^{22} + $$46\!\cdots\!33$$ p^{24} T^{24} + 165402918375811934 p^{30} T^{26} + 467576763291 p^{36} T^{28} + 938962 p^{42} T^{30} + p^{48} T^{32} $
	59	$ 1 - 937618 T^{2} + 432339337947 T^{4} - 128957067202564334 T^{6} + $$29\!\cdots\!01$$ T^{8} - $$61\!\cdots\!08$$ T^{10} + $$14\!\cdots\!42$$ T^{12} - $$10\!\cdots\!40$$ p^{2} T^{14} + $$66\!\cdots\!62$$ p^{4} T^{16} - $$10\!\cdots\!40$$ p^{8} T^{18} + $$14\!\cdots\!42$$ p^{12} T^{20} - $$61\!\cdots\!08$$ p^{18} T^{22} + $$29\!\cdots\!01$$ p^{24} T^{24} - 128957067202564334 p^{30} T^{26} + 432339337947 p^{36} T^{28} - 937618 p^{42} T^{30} + p^{48} T^{32} $
	61	$ ( 1 - 816 T + 1193926 T^{2} - 13002144 p T^{3} + 758515460050 T^{4} - 6819529429512 p T^{5} + 297926214028110592 T^{6} - $$13\!\cdots\!16$$ T^{7} + $$81\!\cdots\!47$$ T^{8} - $$13\!\cdots\!16$$ p^{3} T^{9} + 297926214028110592 p^{6} T^{10} - 6819529429512 p^{10} T^{11} + 758515460050 p^{12} T^{12} - 13002144 p^{16} T^{13} + 1193926 p^{18} T^{14} - 816 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	67	$ ( 1 + 764 T - 273663 T^{2} - 344243228 T^{3} - 11559707059 T^{4} + 21061240190208 T^{5} - 22917938128164926 T^{6} + 4683940411980034808 T^{7} + $$15\!\cdots\!66$$ T^{8} + 4683940411980034808 p^{3} T^{9} - 22917938128164926 p^{6} T^{10} + 21061240190208 p^{9} T^{11} - 11559707059 p^{12} T^{12} - 344243228 p^{15} T^{13} - 273663 p^{18} T^{14} + 764 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	71	$ ( 1 - 1856440 T^{2} + 1761671174068 T^{4} - 1073069689503907240 T^{6} + $$45\!\cdots\!66$$ T^{8} - 1073069689503907240 p^{6} T^{10} + 1761671174068 p^{12} T^{12} - 1856440 p^{18} T^{14} + p^{24} T^{16} )^{2} $
	73	$ ( 1 - 1350 T + 1649863 T^{2} - 1407190050 T^{3} + 1214635307077 T^{4} - 919047647125260 T^{5} + 624120328609647082 T^{6} - $$41\!\cdots\!20$$ T^{7} + $$25\!\cdots\!02$$ T^{8} - $$41\!\cdots\!20$$ p^{3} T^{9} + 624120328609647082 p^{6} T^{10} - 919047647125260 p^{9} T^{11} + 1214635307077 p^{12} T^{12} - 1407190050 p^{15} T^{13} + 1649863 p^{18} T^{14} - 1350 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	79	$ ( 1 + 182 T - 1303458 T^{2} + 51410212 T^{3} + 935956948277 T^{4} - 127720931027844 T^{5} - 463014917975063918 T^{6} + 34428841086426043166 T^{7} + $$20\!\cdots\!88$$ T^{8} + 34428841086426043166 p^{3} T^{9} - 463014917975063918 p^{6} T^{10} - 127720931027844 p^{9} T^{11} + 935956948277 p^{12} T^{12} + 51410212 p^{15} T^{13} - 1303458 p^{18} T^{14} + 182 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	83	$ ( 1 + 2984326 T^{2} + 4434375721357 T^{4} + 4272111895669322434 T^{6} + $$28\!\cdots\!68$$ T^{8} + 4272111895669322434 p^{6} T^{10} + 4434375721357 p^{12} T^{12} + 2984326 p^{18} T^{14} + p^{24} T^{16} )^{2} $
	89	$ 1 - 2563264 T^{2} + 3275474914692 T^{4} - 2650752142645956224 T^{6} + $$12\!\cdots\!98$$ T^{8} + $$65\!\cdots\!92$$ T^{10} - $$73\!\cdots\!16$$ T^{12} + $$87\!\cdots\!44$$ T^{14} - $$70\!\cdots\!05$$ T^{16} + $$87\!\cdots\!44$$ p^{6} T^{18} - $$73\!\cdots\!16$$ p^{12} T^{20} + $$65\!\cdots\!92$$ p^{18} T^{22} + $$12\!\cdots\!98$$ p^{24} T^{24} - 2650752142645956224 p^{30} T^{26} + 3275474914692 p^{36} T^{28} - 2563264 p^{42} T^{30} + p^{48} T^{32} $
	97	$ ( 1 - 2165930 T^{2} + 2100306931633 T^{4} - 316975242262618658 T^{6} - $$53\!\cdots\!72$$ T^{8} - 316975242262618658 p^{6} T^{10} + 2100306931633 p^{12} T^{12} - 2165930 p^{18} T^{14} + p^{24} T^{16} )^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−1.84402629606082249532457925397, −1.81883608636843528159775702868, −1.81407780342717247008407190750, −1.72221519867921360528708385765, −1.66907998270467635003976233638, −1.66448126406684737381528481633, −1.58242251634078591076344351054, −1.43468066596024235559428212192, −1.41918850168401296130606183826, −1.11208551543194347190842860167, −1.03612885797259258776381543436, −0.999201223581665633324522601299, −0.909371453132971143041610003519, −0.872895992626310288291299726034, −0.857974027136214195056905800382, −0.70679213864043055150697903442, −0.61586817050689507837838573214, −0.61586191392768813900852756434, −0.61039354121528913961778899173, −0.56930978941114828214716432564, −0.54201512326477246297748584260, −0.53719540107026925174478833319, −0.44104839799206860300686691549, −0.21099719541794123411754625681, −0.20315737136938805627001332181, 0.20315737136938805627001332181, 0.21099719541794123411754625681, 0.44104839799206860300686691549, 0.53719540107026925174478833319, 0.54201512326477246297748584260, 0.56930978941114828214716432564, 0.61039354121528913961778899173, 0.61586191392768813900852756434, 0.61586817050689507837838573214, 0.70679213864043055150697903442, 0.857974027136214195056905800382, 0.872895992626310288291299726034, 0.909371453132971143041610003519, 0.999201223581665633324522601299, 1.03612885797259258776381543436, 1.11208551543194347190842860167, 1.41918850168401296130606183826, 1.43468066596024235559428212192, 1.58242251634078591076344351054, 1.66448126406684737381528481633, 1.66907998270467635003976233638, 1.72221519867921360528708385765, 1.81407780342717247008407190750, 1.81883608636843528159775702868, 1.84402629606082249532457925397

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1917.067563\)
\(L(\frac12)\)	\(\approx\)	\(1917.067563\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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