LMFDB - L-function 32-117e16-1.1-c3e16-0-0

Dirichlet series

L(s) = 1

+ 12·4^-s + 22·7^-s + 36·13^-s + 101·16^-s − 244·19^-s − 646·25^-s + 264·28^-s + 484·31^-s − 1.01e3·37^-s − 74·43^-s + 1.46e3·49^-s + 432·52^-s − 1.14e3·61^-s + 1.30e3·64^-s + 2.19e3·67^-s − 4.35e3·73^-s − 2.92e3·76^-s + 3.72e3·79^-s + 792·91^-s + 4.37e3·97^-s − 7.75e3·100^-s + 6.90e3·103^-s + 9.70e3·109^-s + 2.22e3·112^-s + 928·121^-s + 5.80e3·124^-s + 127^-s + ⋯

L(s) = 1

+ 3/2·4^-s + 1.18·7^-s + 0.768·13^-s + 1.57·16^-s − 2.94·19^-s − 5.16·25^-s + 1.78·28^-s + 2.80·31^-s − 4.52·37^-s − 0.262·43^-s + 4.27·49^-s + 1.15·52^-s − 2.40·61^-s + 2.54·64^-s + 4.00·67^-s − 6.97·73^-s − 4.41·76^-s + 5.30·79^-s + 0.912·91^-s + 4.57·97^-s − 7.75·100^-s + 6.60·103^-s + 8.53·109^-s + 1.87·112^-s + 0.697·121^-s + 4.20·124^-s + 0.000698·127^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$32$
Conductor:	$3^{32} \cdot 13^{16}$
Sign:	$1$
Analytic conductor:	$2.65968\times 10^{13}$
Root analytic conductor:	$2.62739$
Motivic weight:	$3$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(32,\ 3^{32} \cdot 13^{16} ,\ ( \ : [3/2]^{16} ),\ 1 )$

Particular Values

$L(2)$	$\approx$	$1.113993174$
$L(\frac12)$	$\approx$	$1.113993174$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	13	$ ( 1 - 18 T + 218 p T^{2} - 9036 p T^{3} + 17307 p^{2} T^{4} - 9036 p^{4} T^{5} + 218 p^{7} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8} )^{2} $
good	2	$ 1 - 3 p^{2} T^{2} + 43 T^{4} - 19 p^{5} T^{6} + 5585 T^{8} - 13011 p T^{10} + 146617 p^{2} T^{12} - 70397 p^{6} T^{14} + 47203 p^{8} T^{16} - 70397 p^{12} T^{18} + 146617 p^{14} T^{20} - 13011 p^{19} T^{22} + 5585 p^{24} T^{24} - 19 p^{35} T^{26} + 43 p^{36} T^{28} - 3 p^{44} T^{30} + p^{48} T^{32} $
	5	$ ( 1 + 323 T^{2} + 14109 p T^{4} + 2362538 p T^{6} + 1522988542 T^{8} + 2362538 p^{7} T^{10} + 14109 p^{13} T^{12} + 323 p^{18} T^{14} + p^{24} T^{16} )^{2} $
	7	$ ( 1 - 11 T - 551 T^{2} - 6542 T^{3} + 261099 T^{4} + 3656445 T^{5} + 31833036 T^{6} - 1196598735 T^{7} - 22740718598 T^{8} - 1196598735 p^{3} T^{9} + 31833036 p^{6} T^{10} + 3656445 p^{9} T^{11} + 261099 p^{12} T^{12} - 6542 p^{15} T^{13} - 551 p^{18} T^{14} - 11 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	11	$ 1 - 928 T^{2} - 3785664 T^{4} + 4053892160 T^{6} + 7556560210978 T^{8} - 8001553722822048 T^{10} - 7398746223614361856 T^{12} + $$77\!\cdots\!80$$ T^{14} + $$40\!\cdots\!39$$ T^{16} + $$77\!\cdots\!80$$ p^{6} T^{18} - 7398746223614361856 p^{12} T^{20} - 8001553722822048 p^{18} T^{22} + 7556560210978 p^{24} T^{24} + 4053892160 p^{30} T^{26} - 3785664 p^{36} T^{28} - 928 p^{42} T^{30} + p^{48} T^{32} $
	17	$ 1 - 9879 T^{2} + 30753244 T^{4} + 166586200969 T^{6} - 1660522458491647 T^{8} + 2764817951894364480 T^{10} + $$13\!\cdots\!50$$ T^{12} - $$34\!\cdots\!50$$ T^{14} - $$14\!\cdots\!32$$ T^{16} - $$34\!\cdots\!50$$ p^{6} T^{18} + $$13\!\cdots\!50$$ p^{12} T^{20} + 2764817951894364480 p^{18} T^{22} - 1660522458491647 p^{24} T^{24} + 166586200969 p^{30} T^{26} + 30753244 p^{36} T^{28} - 9879 p^{42} T^{30} + p^{48} T^{32} $
	19	$ ( 1 + 122 T - 4378 T^{2} - 846144 T^{3} - 8602778 T^{4} - 3315450962 T^{5} - 528090317040 T^{6} + 22764820342094 T^{7} + 6522018525318459 T^{8} + 22764820342094 p^{3} T^{9} - 528090317040 p^{6} T^{10} - 3315450962 p^{9} T^{11} - 8602778 p^{12} T^{12} - 846144 p^{15} T^{13} - 4378 p^{18} T^{14} + 122 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	23	$ 1 - 50984 T^{2} + 1415091248 T^{4} - 25458278746576 T^{6} + 298022901417174050 T^{8} - $$14\!\cdots\!80$$ T^{10} - $$28\!\cdots\!64$$ T^{12} + $$89\!\cdots\!08$$ T^{14} - $$13\!\cdots\!89$$ T^{16} + $$89\!\cdots\!08$$ p^{6} T^{18} - $$28\!\cdots\!64$$ p^{12} T^{20} - $$14\!\cdots\!80$$ p^{18} T^{22} + 298022901417174050 p^{24} T^{24} - 25458278746576 p^{30} T^{26} + 1415091248 p^{36} T^{28} - 50984 p^{42} T^{30} + p^{48} T^{32} $
	29	$ 1 - 55747 T^{2} + 418356456 T^{4} + 46071545329505 T^{6} - 723389975882976467 T^{8} - $$43\!\cdots\!72$$ T^{10} + $$15\!\cdots\!14$$ T^{12} + $$44\!\cdots\!90$$ T^{14} - $$87\!\cdots\!36$$ T^{16} + $$44\!\cdots\!90$$ p^{6} T^{18} + $$15\!\cdots\!14$$ p^{12} T^{20} - $$43\!\cdots\!72$$ p^{18} T^{22} - 723389975882976467 p^{24} T^{24} + 46071545329505 p^{30} T^{26} + 418356456 p^{36} T^{28} - 55747 p^{42} T^{30} + p^{48} T^{32} $
	31	$ ( 1 - 121 T + 64928 T^{2} - 1397453 T^{3} + 1743770334 T^{4} - 1397453 p^{3} T^{5} + 64928 p^{6} T^{6} - 121 p^{9} T^{7} + p^{12} T^{8} )^{4} $
	37	$ ( 1 + 509 T + 26318 T^{2} - 9398401 T^{3} + 4364634275 T^{4} + 540168859860 T^{5} - 385838743905372 T^{6} - 33344367024333064 T^{7} + 12929640021335095148 T^{8} - 33344367024333064 p^{3} T^{9} - 385838743905372 p^{6} T^{10} + 540168859860 p^{9} T^{11} + 4364634275 p^{12} T^{12} - 9398401 p^{15} T^{13} + 26318 p^{18} T^{14} + 509 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	41	$ 1 - 325831 T^{2} + 1237523692 p T^{4} - 5629725002461271 T^{6} + $$54\!\cdots\!45$$ T^{8} - $$48\!\cdots\!76$$ T^{10} + $$38\!\cdots\!62$$ T^{12} - $$27\!\cdots\!46$$ T^{14} + $$19\!\cdots\!68$$ T^{16} - $$27\!\cdots\!46$$ p^{6} T^{18} + $$38\!\cdots\!62$$ p^{12} T^{20} - $$48\!\cdots\!76$$ p^{18} T^{22} + $$54\!\cdots\!45$$ p^{24} T^{24} - 5629725002461271 p^{30} T^{26} + 1237523692 p^{37} T^{28} - 325831 p^{42} T^{30} + p^{48} T^{32} $
	43	$ ( 1 + 37 T - 170487 T^{2} + 7519730 T^{3} + 13547589531 T^{4} - 1330543244599 T^{5} - 576362129219728 T^{6} + 65362487105178093 T^{7} + 30959535359997330566 T^{8} + 65362487105178093 p^{3} T^{9} - 576362129219728 p^{6} T^{10} - 1330543244599 p^{9} T^{11} + 13547589531 p^{12} T^{12} + 7519730 p^{15} T^{13} - 170487 p^{18} T^{14} + 37 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	47	$ ( 1 + 673128 T^{2} + 205725969680 T^{4} + 38035116822467736 T^{6} + $$47\!\cdots\!38$$ T^{8} + 38035116822467736 p^{6} T^{10} + 205725969680 p^{12} T^{12} + 673128 p^{18} T^{14} + p^{24} T^{16} )^{2} $
	53	$ ( 1 + 457499 T^{2} + 135151726097 T^{4} + 29356704635913714 T^{6} + $$48\!\cdots\!10$$ T^{8} + 29356704635913714 p^{6} T^{10} + 135151726097 p^{12} T^{12} + 457499 p^{18} T^{14} + p^{24} T^{16} )^{2} $
	59	$ 1 - 1197976 T^{2} + 776298092804 T^{4} - 339993714292870352 T^{6} + $$11\!\cdots\!94$$ T^{8} - $$28\!\cdots\!20$$ T^{10} + $$59\!\cdots\!44$$ T^{12} - $$11\!\cdots\!88$$ T^{14} + $$21\!\cdots\!27$$ T^{16} - $$11\!\cdots\!88$$ p^{6} T^{18} + $$59\!\cdots\!44$$ p^{12} T^{20} - $$28\!\cdots\!20$$ p^{18} T^{22} + $$11\!\cdots\!94$$ p^{24} T^{24} - 339993714292870352 p^{30} T^{26} + 776298092804 p^{36} T^{28} - 1197976 p^{42} T^{30} + p^{48} T^{32} $
	61	$ ( 1 + 574 T - 507754 T^{2} - 251650676 T^{3} + 213092347349 T^{4} + 70802544658176 T^{5} - 59904675581544318 T^{6} - 6002083932471028274 T^{7} + $$15\!\cdots\!88$$ T^{8} - 6002083932471028274 p^{3} T^{9} - 59904675581544318 p^{6} T^{10} + 70802544658176 p^{9} T^{11} + 213092347349 p^{12} T^{12} - 251650676 p^{15} T^{13} - 507754 p^{18} T^{14} + 574 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	67	$ ( 1 - 1099 T - 92815 T^{2} + 450750458 T^{3} + 54333876755 T^{4} - 177502816309503 T^{5} + 22544106508715520 T^{6} + 8335536027317263325 T^{7} + $$37\!\cdots\!70$$ T^{8} + 8335536027317263325 p^{3} T^{9} + 22544106508715520 p^{6} T^{10} - 177502816309503 p^{9} T^{11} + 54333876755 p^{12} T^{12} + 450750458 p^{15} T^{13} - 92815 p^{18} T^{14} - 1099 p^{21} T^{15} + p^{24} T^{16} )^{2} $
	71	$ 1 - 1071656 T^{2} + 400059941552 T^{4} - 60778262004222928 T^{6} + $$11\!\cdots\!30$$ T^{8} - $$19\!\cdots\!64$$ T^{10} - $$23\!\cdots\!88$$ T^{12} - $$48\!\cdots\!40$$ T^{14} + $$71\!\cdots\!95$$ T^{16} - $$48\!\cdots\!40$$ p^{6} T^{18} - $$23\!\cdots\!88$$ p^{12} T^{20} - $$19\!\cdots\!64$$ p^{18} T^{22} + $$11\!\cdots\!30$$ p^{24} T^{24} - 60778262004222928 p^{30} T^{26} + 400059941552 p^{36} T^{28} - 1071656 p^{42} T^{30} + p^{48} T^{32} $
	73	$ ( 1 + 1088 T + 1503042 T^{2} + 1067919104 T^{3} + 887747129507 T^{4} + 1067919104 p^{3} T^{5} + 1503042 p^{6} T^{6} + 1088 p^{9} T^{7} + p^{12} T^{8} )^{4} $
	79	$ ( 1 - 931 T + 2149008 T^{2} - 1338612247 T^{3} + 1623052949342 T^{4} - 1338612247 p^{3} T^{5} + 2149008 p^{6} T^{6} - 931 p^{9} T^{7} + p^{12} T^{8} )^{4} $
	83	$ ( 1 + 3704064 T^{2} + 6275226483968 T^{4} + 6440475301192582208 T^{6} + $$44\!\cdots\!02$$ T^{8} + 6440475301192582208 p^{6} T^{10} + 6275226483968 p^{12} T^{12} + 3704064 p^{18} T^{14} + p^{24} T^{16} )^{2} $
	89	$ 1 - 2583560 T^{2} + 2698434804196 T^{4} - 22772105355352304 p T^{6} + $$18\!\cdots\!82$$ T^{8} - $$12\!\cdots\!00$$ T^{10} + $$47\!\cdots\!56$$ T^{12} - $$24\!\cdots\!92$$ T^{14} + $$23\!\cdots\!15$$ T^{16} - $$24\!\cdots\!92$$ p^{6} T^{18} + $$47\!\cdots\!56$$ p^{12} T^{20} - $$12\!\cdots\!00$$ p^{18} T^{22} + $$18\!\cdots\!82$$ p^{24} T^{24} - 22772105355352304 p^{31} T^{26} + 2698434804196 p^{36} T^{28} - 2583560 p^{42} T^{30} + p^{48} T^{32} $
	97	$ ( 1 - 2185 T + 1282907 T^{2} + 1169604368 T^{3} - 2421090742105 T^{4} + 2469689407192071 T^{5} - 1055556593599297002 T^{6} - $$19\!\cdots\!29$$ T^{7} + $$37\!\cdots\!36$$ T^{8} - $$19\!\cdots\!29$$ p^{3} T^{9} - 1055556593599297002 p^{6} T^{10} + 2469689407192071 p^{9} T^{11} - 2421090742105 p^{12} T^{12} + 1169604368 p^{15} T^{13} + 1282907 p^{18} T^{14} - 2185 p^{21} T^{15} + 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

−3.61132804842118143025841401728, −3.33071030426703275580007886966, −3.31199662251078439709500016483, −3.26790258654181164873678307969, −3.21114333251590334347196144927, −2.94848235936147663030206406582, −2.61875130199682089418155576767, −2.57010953556959715715978230001, −2.51144294686675963981910488213, −2.33624065763265132797319037002, −2.19242692947708336355548670218, −2.15070516844580743729055650999, −2.08864406238712216236794757871, −2.05351089065379083337985209717, −1.94622504409769763197675964336, −1.81911609205266016828681485425, −1.55909587933586545232878417688, −1.53152520791604854926011197078, −1.21052035283430772864006123439, −1.05891417195810469445218668742, −0.903282662158869739723650622829, −0.72824653268571087696808694689, −0.62472171171862027855047680792, −0.14279347358145494278169914010, −0.10571199121443442236040288347, 0.10571199121443442236040288347, 0.14279347358145494278169914010, 0.62472171171862027855047680792, 0.72824653268571087696808694689, 0.903282662158869739723650622829, 1.05891417195810469445218668742, 1.21052035283430772864006123439, 1.53152520791604854926011197078, 1.55909587933586545232878417688, 1.81911609205266016828681485425, 1.94622504409769763197675964336, 2.05351089065379083337985209717, 2.08864406238712216236794757871, 2.15070516844580743729055650999, 2.19242692947708336355548670218, 2.33624065763265132797319037002, 2.51144294686675963981910488213, 2.57010953556959715715978230001, 2.61875130199682089418155576767, 2.94848235936147663030206406582, 3.21114333251590334347196144927, 3.26790258654181164873678307969, 3.31199662251078439709500016483, 3.33071030426703275580007886966, 3.61132804842118143025841401728

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.113993174\)
\(L(\frac12)\)	\(\approx\)	\(1.113993174\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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