LMFDB - L-function 40-14e40-1.1-c3e20-0-1

Dirichlet series

L(s) = 1

+ 2·4^-s + 6·5^-s + 24·8^-s + 107·9^-s − 24·16^-s + 6·17^-s + 12·20^-s − 625·25^-s − 352·29^-s + 40·32^-s + 214·36^-s + 258·37^-s + 144·40^-s + 642·45^-s + 570·53^-s − 294·61^-s + 664·64^-s + 12·68^-s + 2.56e3·72^-s − 966·73^-s − 144·80^-s + 5.78e3·81^-s + 36·85^-s + 3.18e3·89^-s − 1.25e3·100^-s − 2.70e3·101^-s − 114·109^-s + ⋯

L(s) = 1

+ 1/4·4^-s + 0.536·5^-s + 1.06·8^-s + 3.96·9^-s − 3/8·16^-s + 0.0856·17^-s + 0.134·20^-s − 5·25^-s − 2.25·29^-s + 0.220·32^-s + 0.990·36^-s + 1.14·37^-s + 0.569·40^-s + 2.12·45^-s + 1.47·53^-s − 0.617·61^-s + 1.29·64^-s + 0.0214·68^-s + 4.20·72^-s − 1.54·73^-s − 0.201·80^-s + 7.93·81^-s + 0.0459·85^-s + 3.79·89^-s − 5/4·100^-s − 2.66·101^-s − 0.100·109^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{40}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 7^{40}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$40$
Conductor:	$2^{40} \cdot 7^{40}$
Sign:	$1$
Analytic conductor:	$1.82996\times 10^{21}$
Root analytic conductor:	$3.40064$
Motivic weight:	$3$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(40,\ 2^{40} \cdot 7^{40} ,\ ( \ : [3/2]^{20} ),\ 1 )$

Particular Values

$L(2)$	$\approx$	$95.56972392$
$L(\frac12)$	$\approx$	$95.56972392$
$L(\frac{5}{2})$		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 - p T^{2} - 3 p^{3} T^{3} + 7 p^{2} T^{4} + 7 p^{3} T^{5} - 3 p^{6} T^{6} + 11 p^{5} T^{7} - 7 p^{6} T^{8} + 21 p^{8} T^{9} - 81 p^{9} T^{10} + 21 p^{11} T^{11} - 7 p^{12} T^{12} + 11 p^{14} T^{13} - 3 p^{18} T^{14} + 7 p^{18} T^{15} + 7 p^{20} T^{16} - 3 p^{24} T^{17} - p^{25} T^{18} + p^{30} T^{20} $
	7	$ 1 $
good	3	$ 1 - 107 T^{2} + 5662 T^{4} - 187613 T^{6} + 3831154 T^{8} - 26110181 T^{10} - 1423228964 T^{12} + 71438208067 T^{14} - 1719547265195 T^{16} + 7570956216478 p T^{18} - 29854652081252 p^{2} T^{20} + 7570956216478 p^{7} T^{22} - 1719547265195 p^{12} T^{24} + 71438208067 p^{18} T^{26} - 1423228964 p^{24} T^{28} - 26110181 p^{30} T^{30} + 3831154 p^{36} T^{32} - 187613 p^{42} T^{34} + 5662 p^{48} T^{36} - 107 p^{54} T^{38} + p^{60} T^{40} $
	5	$ ( 1 - 3 T + 326 T^{2} - 969 T^{3} + 56386 T^{4} - 285909 T^{5} + 5130156 T^{6} - 65106273 T^{7} + 195537701 T^{8} - 2167612206 p T^{9} - 378529748 p^{2} T^{10} - 2167612206 p^{4} T^{11} + 195537701 p^{6} T^{12} - 65106273 p^{9} T^{13} + 5130156 p^{12} T^{14} - 285909 p^{15} T^{15} + 56386 p^{18} T^{16} - 969 p^{21} T^{17} + 326 p^{24} T^{18} - 3 p^{27} T^{19} + p^{30} T^{20} )^{2} $
	11	$ 1 + 7325 T^{2} + 24359934 T^{4} + 55975723019 T^{6} + 10868425555606 p T^{8} + 251719854045201651 T^{10} + $$47\!\cdots\!24$$ T^{12} + $$77\!\cdots\!87$$ T^{14} + $$11\!\cdots\!05$$ T^{16} + $$17\!\cdots\!38$$ T^{18} + $$25\!\cdots\!16$$ T^{20} + $$17\!\cdots\!38$$ p^{6} T^{22} + $$11\!\cdots\!05$$ p^{12} T^{24} + $$77\!\cdots\!87$$ p^{18} T^{26} + $$47\!\cdots\!24$$ p^{24} T^{28} + 251719854045201651 p^{30} T^{30} + 10868425555606 p^{37} T^{32} + 55975723019 p^{42} T^{34} + 24359934 p^{48} T^{36} + 7325 p^{54} T^{38} + p^{60} T^{40} $
	13	$ ( 1 - 16046 T^{2} + 123630613 T^{4} - 604920720264 T^{6} + 2086415842231586 T^{8} - 5304798739224059092 T^{10} + 2086415842231586 p^{6} T^{12} - 604920720264 p^{12} T^{14} + 123630613 p^{18} T^{16} - 16046 p^{24} T^{18} + p^{30} T^{20} )^{2} $
	17	$ ( 1 - 3 T + 13842 T^{2} - 41517 T^{3} + 103633626 T^{4} + 3058334115 T^{5} + 515984790576 T^{6} + 44911391158707 T^{7} + 1569369672025053 T^{8} + 342003710929684410 T^{9} + 5092133157312142604 T^{10} + 342003710929684410 p^{3} T^{11} + 1569369672025053 p^{6} T^{12} + 44911391158707 p^{9} T^{13} + 515984790576 p^{12} T^{14} + 3058334115 p^{15} T^{15} + 103633626 p^{18} T^{16} - 41517 p^{21} T^{17} + 13842 p^{24} T^{18} - 3 p^{27} T^{19} + p^{30} T^{20} )^{2} $
	19	$ 1 - 33923 T^{2} + 473473550 T^{4} - 4739803420581 T^{6} + 57017326480335890 T^{8} - $$62\!\cdots\!69$$ T^{10} + $$49\!\cdots\!68$$ T^{12} - $$40\!\cdots\!53$$ T^{14} + $$36\!\cdots\!93$$ T^{16} - $$26\!\cdots\!94$$ T^{18} + $$16\!\cdots\!00$$ T^{20} - $$26\!\cdots\!94$$ p^{6} T^{22} + $$36\!\cdots\!93$$ p^{12} T^{24} - $$40\!\cdots\!53$$ p^{18} T^{26} + $$49\!\cdots\!68$$ p^{24} T^{28} - $$62\!\cdots\!69$$ p^{30} T^{30} + 57017326480335890 p^{36} T^{32} - 4739803420581 p^{42} T^{34} + 473473550 p^{48} T^{36} - 33923 p^{54} T^{38} + p^{60} T^{40} $
	23	$ 1 + 82949 T^{2} + 3683699478 T^{4} + 110940145241627 T^{6} + 2486122768063289954 T^{8} + $$42\!\cdots\!23$$ T^{10} + $$56\!\cdots\!48$$ T^{12} + $$53\!\cdots\!67$$ T^{14} + $$27\!\cdots\!13$$ T^{16} - $$12\!\cdots\!30$$ T^{18} - $$39\!\cdots\!84$$ T^{20} - $$12\!\cdots\!30$$ p^{6} T^{22} + $$27\!\cdots\!13$$ p^{12} T^{24} + $$53\!\cdots\!67$$ p^{18} T^{26} + $$56\!\cdots\!48$$ p^{24} T^{28} + $$42\!\cdots\!23$$ p^{30} T^{30} + 2486122768063289954 p^{36} T^{32} + 110940145241627 p^{42} T^{34} + 3683699478 p^{48} T^{36} + 82949 p^{54} T^{38} + p^{60} T^{40} $
	29	$ ( 1 + 88 T + 81149 T^{2} + 3661360 T^{3} + 2960904150 T^{4} + 78971527600 T^{5} + 2960904150 p^{3} T^{6} + 3661360 p^{6} T^{7} + 81149 p^{9} T^{8} + 88 p^{12} T^{9} + p^{15} T^{10} )^{4} $
	31	$ 1 - 168499 T^{2} + 17030998774 T^{4} - 1212483426198141 T^{6} + 67276661467762565922 T^{8} - $$30\!\cdots\!05$$ T^{10} + $$11\!\cdots\!88$$ T^{12} - $$34\!\cdots\!73$$ T^{14} + $$94\!\cdots\!37$$ T^{16} - $$23\!\cdots\!42$$ T^{18} + $$65\!\cdots\!20$$ T^{20} - $$23\!\cdots\!42$$ p^{6} T^{22} + $$94\!\cdots\!37$$ p^{12} T^{24} - $$34\!\cdots\!73$$ p^{18} T^{26} + $$11\!\cdots\!88$$ p^{24} T^{28} - $$30\!\cdots\!05$$ p^{30} T^{30} + 67276661467762565922 p^{36} T^{32} - 1212483426198141 p^{42} T^{34} + 17030998774 p^{48} T^{36} - 168499 p^{54} T^{38} + p^{60} T^{40} $
	37	$ ( 1 - 129 T - 206026 T^{2} + 14977281 T^{3} + 25629664526 T^{4} - 986927645919 T^{5} - 2290168598305336 T^{6} + 44884976449668597 T^{7} + $$15\!\cdots\!45$$ T^{8} - $$99\!\cdots\!94$$ T^{9} - $$89\!\cdots\!04$$ T^{10} - $$99\!\cdots\!94$$ p^{3} T^{11} + $$15\!\cdots\!45$$ p^{6} T^{12} + 44884976449668597 p^{9} T^{13} - 2290168598305336 p^{12} T^{14} - 986927645919 p^{15} T^{15} + 25629664526 p^{18} T^{16} + 14977281 p^{21} T^{17} - 206026 p^{24} T^{18} - 129 p^{27} T^{19} + p^{30} T^{20} )^{2} $
	41	$ ( 1 - 318166 T^{2} + 54625842349 T^{4} - 6625411048182696 T^{6} + $$62\!\cdots\!10$$ T^{8} - $$47\!\cdots\!64$$ T^{10} + $$62\!\cdots\!10$$ p^{6} T^{12} - 6625411048182696 p^{12} T^{14} + 54625842349 p^{18} T^{16} - 318166 p^{24} T^{18} + p^{30} T^{20} )^{2} $
	43	$ ( 1 - 313882 T^{2} + 54571176181 T^{4} - 6720097962314232 T^{6} + $$66\!\cdots\!46$$ T^{8} - $$56\!\cdots\!68$$ T^{10} + $$66\!\cdots\!46$$ p^{6} T^{12} - 6720097962314232 p^{12} T^{14} + 54571176181 p^{18} T^{16} - 313882 p^{24} T^{18} + p^{30} T^{20} )^{2} $
	47	$ 1 - 489659 T^{2} + 138839571590 T^{4} - 27002559990918805 T^{6} + $$39\!\cdots\!74$$ T^{8} - $$45\!\cdots\!05$$ T^{10} + $$43\!\cdots\!32$$ T^{12} - $$38\!\cdots\!09$$ T^{14} + $$35\!\cdots\!17$$ T^{16} - $$37\!\cdots\!26$$ T^{18} + $$39\!\cdots\!80$$ T^{20} - $$37\!\cdots\!26$$ p^{6} T^{22} + $$35\!\cdots\!17$$ p^{12} T^{24} - $$38\!\cdots\!09$$ p^{18} T^{26} + $$43\!\cdots\!32$$ p^{24} T^{28} - $$45\!\cdots\!05$$ p^{30} T^{30} + $$39\!\cdots\!74$$ p^{36} T^{32} - 27002559990918805 p^{42} T^{34} + 138839571590 p^{48} T^{36} - 489659 p^{54} T^{38} + p^{60} T^{40} $
	53	$ ( 1 - 285 T - 358946 T^{2} + 134381277 T^{3} + 59282987854 T^{4} - 29221778327395 T^{5} - 4859397253553088 T^{6} + 4605660238118684977 T^{7} - $$28\!\cdots\!47$$ T^{8} - $$31\!\cdots\!34$$ T^{9} + $$13\!\cdots\!28$$ T^{10} - $$31\!\cdots\!34$$ p^{3} T^{11} - $$28\!\cdots\!47$$ p^{6} T^{12} + 4605660238118684977 p^{9} T^{13} - 4859397253553088 p^{12} T^{14} - 29221778327395 p^{15} T^{15} + 59282987854 p^{18} T^{16} + 134381277 p^{21} T^{17} - 358946 p^{24} T^{18} - 285 p^{27} T^{19} + p^{30} T^{20} )^{2} $
	59	$ 1 - 1238899 T^{2} + 744290665598 T^{4} - 309086890380590085 T^{6} + $$10\!\cdots\!22$$ T^{8} - $$32\!\cdots\!25$$ T^{10} + $$90\!\cdots\!04$$ T^{12} - $$23\!\cdots\!05$$ T^{14} + $$55\!\cdots\!49$$ T^{16} - $$12\!\cdots\!14$$ T^{18} + $$26\!\cdots\!32$$ T^{20} - $$12\!\cdots\!14$$ p^{6} T^{22} + $$55\!\cdots\!49$$ p^{12} T^{24} - $$23\!\cdots\!05$$ p^{18} T^{26} + $$90\!\cdots\!04$$ p^{24} T^{28} - $$32\!\cdots\!25$$ p^{30} T^{30} + $$10\!\cdots\!22$$ p^{36} T^{32} - 309086890380590085 p^{42} T^{34} + 744290665598 p^{48} T^{36} - 1238899 p^{54} T^{38} + p^{60} T^{40} $
	61	$ ( 1 + 147 T + 512650 T^{2} + 74300709 T^{3} + 97298273830 T^{4} - 17932548454947 T^{5} + 13357089464161932 T^{6} - 15848238642795148119 T^{7} + $$17\!\cdots\!41$$ T^{8} - $$35\!\cdots\!62$$ T^{9} + $$15\!\cdots\!08$$ T^{10} - $$35\!\cdots\!62$$ p^{3} T^{11} + $$17\!\cdots\!41$$ p^{6} T^{12} - 15848238642795148119 p^{9} T^{13} + 13357089464161932 p^{12} T^{14} - 17932548454947 p^{15} T^{15} + 97298273830 p^{18} T^{16} + 74300709 p^{21} T^{17} + 512650 p^{24} T^{18} + 147 p^{27} T^{19} + p^{30} T^{20} )^{2} $
	67	$ 1 + 1184021 T^{2} + 507383288094 T^{4} + 109298906755516067 T^{6} + $$40\!\cdots\!78$$ T^{8} + $$22\!\cdots\!07$$ T^{10} + $$66\!\cdots\!12$$ T^{12} + $$10\!\cdots\!63$$ T^{14} + $$31\!\cdots\!45$$ T^{16} + $$12\!\cdots\!90$$ T^{18} + $$41\!\cdots\!84$$ T^{20} + $$12\!\cdots\!90$$ p^{6} T^{22} + $$31\!\cdots\!45$$ p^{12} T^{24} + $$10\!\cdots\!63$$ p^{18} T^{26} + $$66\!\cdots\!12$$ p^{24} T^{28} + $$22\!\cdots\!07$$ p^{30} T^{30} + $$40\!\cdots\!78$$ p^{36} T^{32} + 109298906755516067 p^{42} T^{34} + 507383288094 p^{48} T^{36} + 1184021 p^{54} T^{38} + p^{60} T^{40} $
	71	$ ( 1 - 1948194 T^{2} + 1444984783357 T^{4} - 382327360856464920 T^{6} - $$99\!\cdots\!86$$ T^{8} + $$88\!\cdots\!04$$ T^{10} - $$99\!\cdots\!86$$ p^{6} T^{12} - 382327360856464920 p^{12} T^{14} + 1444984783357 p^{18} T^{16} - 1948194 p^{24} T^{18} + p^{30} T^{20} )^{2} $
	73	$ ( 1 + 483 T + 941022 T^{2} + 416954097 T^{3} + 256426955766 T^{4} + 61516121290869 T^{5} + 43061585257716864 T^{6} + 9555800747270060085 T^{7} + $$59\!\cdots\!93$$ T^{8} + $$37\!\cdots\!54$$ T^{9} + $$38\!\cdots\!20$$ T^{10} + $$37\!\cdots\!54$$ p^{3} T^{11} + $$59\!\cdots\!93$$ p^{6} T^{12} + 9555800747270060085 p^{9} T^{13} + 43061585257716864 p^{12} T^{14} + 61516121290869 p^{15} T^{15} + 256426955766 p^{18} T^{16} + 416954097 p^{21} T^{17} + 941022 p^{24} T^{18} + 483 p^{27} T^{19} + p^{30} T^{20} )^{2} $
	79	$ 1 + 2565797 T^{2} + 3035106686406 T^{4} + 2384091019089619979 T^{6} + $$15\!\cdots\!82$$ T^{8} + $$96\!\cdots\!79$$ T^{10} + $$55\!\cdots\!64$$ T^{12} + $$30\!\cdots\!35$$ T^{14} + $$16\!\cdots\!85$$ T^{16} + $$86\!\cdots\!30$$ T^{18} + $$42\!\cdots\!48$$ T^{20} + $$86\!\cdots\!30$$ p^{6} T^{22} + $$16\!\cdots\!85$$ p^{12} T^{24} + $$30\!\cdots\!35$$ p^{18} T^{26} + $$55\!\cdots\!64$$ p^{24} T^{28} + $$96\!\cdots\!79$$ p^{30} T^{30} + $$15\!\cdots\!82$$ p^{36} T^{32} + 2384091019089619979 p^{42} T^{34} + 3035106686406 p^{48} T^{36} + 2565797 p^{54} T^{38} + p^{60} T^{40} $
	83	$ ( 1 + 3788174 T^{2} + 7268549244485 T^{4} + 9055923896900693352 T^{6} + $$80\!\cdots\!78$$ T^{8} + $$53\!\cdots\!84$$ T^{10} + $$80\!\cdots\!78$$ p^{6} T^{12} + 9055923896900693352 p^{12} T^{14} + 7268549244485 p^{18} T^{16} + 3788174 p^{24} T^{18} + p^{30} T^{20} )^{2} $
	89	$ ( 1 - 1593 T + 3980966 T^{2} - 4994187219 T^{3} + 7728801075910 T^{4} - 7948331499927375 T^{5} + 9613102812298347624 T^{6} - $$85\!\cdots\!19$$ T^{7} + $$88\!\cdots\!77$$ T^{8} - $$71\!\cdots\!46$$ T^{9} + $$67\!\cdots\!40$$ T^{10} - $$71\!\cdots\!46$$ p^{3} T^{11} + $$88\!\cdots\!77$$ p^{6} T^{12} - $$85\!\cdots\!19$$ p^{9} T^{13} + 9613102812298347624 p^{12} T^{14} - 7948331499927375 p^{15} T^{15} + 7728801075910 p^{18} T^{16} - 4994187219 p^{21} T^{17} + 3980966 p^{24} T^{18} - 1593 p^{27} T^{19} + p^{30} T^{20} )^{2} $
	97	$ ( 1 - 6928838 T^{2} + 22880908134397 T^{4} - 47527912576344311400 T^{6} + $$68\!\cdots\!98$$ T^{8} - $$73\!\cdots\!76$$ T^{10} + $$68\!\cdots\!98$$ p^{6} T^{12} - 47527912576344311400 p^{12} T^{14} + 22880908134397 p^{18} T^{16} - 6928838 p^{24} T^{18} + p^{30} T^{20} )^{2} $
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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

−2.49072005260317376694715059801, −2.38657896112361551239307308515, −2.25968093197271727326445028679, −2.19398495111562223031966327074, −2.13649867009586434562381542587, −2.11462199943454209699564915475, −2.08768340120580413455323597089, −1.90068688296140044284172408256, −1.79384798263890268334746621198, −1.77569335919797181936607867424, −1.64696992500243498939678066983, −1.59095607961546812579871286320, −1.47677649929838588580441041020, −1.34942606685199715182502810939, −1.32281060471031314468080183636, −1.24830459811102107371031969825, −1.22302082290219715195464430279, −1.04062922922169776607968222254, −0.795905270749473404760572321012, −0.56506985861578713479281154538, −0.55826223611953109982104788627, −0.52049948399408894154968048005, −0.30663115547712812901349301506, −0.28057742958226821248796981539, −0.26224490003351297567838345411, 0.26224490003351297567838345411, 0.28057742958226821248796981539, 0.30663115547712812901349301506, 0.52049948399408894154968048005, 0.55826223611953109982104788627, 0.56506985861578713479281154538, 0.795905270749473404760572321012, 1.04062922922169776607968222254, 1.22302082290219715195464430279, 1.24830459811102107371031969825, 1.32281060471031314468080183636, 1.34942606685199715182502810939, 1.47677649929838588580441041020, 1.59095607961546812579871286320, 1.64696992500243498939678066983, 1.77569335919797181936607867424, 1.79384798263890268334746621198, 1.90068688296140044284172408256, 2.08768340120580413455323597089, 2.11462199943454209699564915475, 2.13649867009586434562381542587, 2.19398495111562223031966327074, 2.25968093197271727326445028679, 2.38657896112361551239307308515, 2.49072005260317376694715059801

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(95.56972392\)
\(L(\frac12)\)	\(\approx\)	\(95.56972392\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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