LMFDB - L-function 10-304e5-1.1-c7e5-0-0

Dirichlet series

L(s) = 1

+ 14·3^-s − 280·5^-s − 414·7^-s − 3.48e3·9^-s + 2.66e3·11^-s − 602·13^-s − 3.92e3·15^-s − 2.73e4·17^-s + 3.42e4·19^-s − 5.79e3·21^-s + 6.70e4·23^-s − 2.10e5·25^-s − 248·27^-s − 3.72e5·29^-s + 2.71e5·31^-s + 3.72e4·33^-s + 1.15e5·35^-s − 5.62e5·37^-s − 8.42e3·39^-s − 9.56e5·41^-s + 8.27e5·43^-s + 9.74e5·45^-s + 1.81e6·47^-s − 2.40e6·49^-s − 3.83e5·51^-s + 4.86e5·53^-s − 7.45e5·55^-s + ⋯

L(s) = 1

+ 0.299·3^-s − 1.00·5^-s − 0.456·7^-s − 1.59·9^-s + 0.603·11^-s − 0.0759·13^-s − 0.299·15^-s − 1.35·17^-s + 1.14·19^-s − 0.136·21^-s + 1.14·23^-s − 2.69·25^-s − 0.00242·27^-s − 2.83·29^-s + 1.63·31^-s + 0.180·33^-s + 0.457·35^-s − 1.82·37^-s − 0.0227·39^-s − 2.16·41^-s + 1.58·43^-s + 1.59·45^-s + 2.54·47^-s − 2.91·49^-s − 0.404·51^-s + 0.449·53^-s − 0.604·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$10$
Conductor:	$2^{20} \cdot 19^{5}$
Sign:	$1$
Analytic conductor:	$7.72358\times 10^{9}$
Root analytic conductor:	$9.74500$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(10,\ 2^{20} \cdot 19^{5} ,\ ( \ : 7/2, 7/2, 7/2, 7/2, 7/2 ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$2.702338824$
$L(\frac12)$	$\approx$	$2.702338824$
$L(\frac{9}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$\Gal(F_p)$	$F_p(T)$
bad	2		$ 1 $
	19	$C_1$	$ ( 1 - p^{3} T )^{5} $
good	3	$C_2 \wr S_5$	$ 1 - 14 T + 3676 T^{2} - 11104 p^{2} T^{3} + 247145 p^{3} T^{4} - 9484444 p^{3} T^{5} + 247145 p^{10} T^{6} - 11104 p^{16} T^{7} + 3676 p^{21} T^{8} - 14 p^{28} T^{9} + p^{35} T^{10} $
	5	$C_2 \wr S_5$	$ 1 + 56 p T + 57814 p T^{2} + 2652558 p^{2} T^{3} + 1563182069 p^{2} T^{4} + 55600139212 p^{3} T^{5} + 1563182069 p^{9} T^{6} + 2652558 p^{16} T^{7} + 57814 p^{22} T^{8} + 56 p^{29} T^{9} + p^{35} T^{10} $
	7	$C_2 \wr S_5$	$ 1 + 414 T + 2575571 T^{2} + 2009518362 T^{3} + 2876468268073 T^{4} + 2752968688370496 T^{5} + 2876468268073 p^{7} T^{6} + 2009518362 p^{14} T^{7} + 2575571 p^{21} T^{8} + 414 p^{28} T^{9} + p^{35} T^{10} $
	11	$C_2 \wr S_5$	$ 1 - 2 p^{3} T + 47532404 T^{2} - 196653658836 T^{3} + 1564587459786083 T^{4} - 4588703502231464828 T^{5} + 1564587459786083 p^{7} T^{6} - 196653658836 p^{14} T^{7} + 47532404 p^{21} T^{8} - 2 p^{31} T^{9} + p^{35} T^{10} $
	13	$C_2 \wr S_5$	$ 1 + 602 T + 147508536 T^{2} + 786462907278 T^{3} + 9885416966268075 T^{4} + 88572120386284541784 T^{5} + 9885416966268075 p^{7} T^{6} + 786462907278 p^{14} T^{7} + 147508536 p^{21} T^{8} + 602 p^{28} T^{9} + p^{35} T^{10} $
	17	$C_2 \wr S_5$	$ 1 + 27366 T + 1416771931 T^{2} + 30995036043348 T^{3} + 976315396052141173 T^{4} + $$16\!\cdots\!54$$ T^{5} + 976315396052141173 p^{7} T^{6} + 30995036043348 p^{14} T^{7} + 1416771931 p^{21} T^{8} + 27366 p^{28} T^{9} + p^{35} T^{10} $
	23	$C_2 \wr S_5$	$ 1 - 67096 T + 672690382 p T^{2} - 742746864613344 T^{3} + 97403622365062360565 T^{4} - $$15\!\cdots\!64$$ p T^{5} + 97403622365062360565 p^{7} T^{6} - 742746864613344 p^{14} T^{7} + 672690382 p^{22} T^{8} - 67096 p^{28} T^{9} + p^{35} T^{10} $
	29	$C_2 \wr S_5$	$ 1 + 372398 T + 123594633620 T^{2} + 24322905221765418 T^{3} + $$45\!\cdots\!35$$ T^{4} + $$60\!\cdots\!48$$ T^{5} + $$45\!\cdots\!35$$ p^{7} T^{6} + 24322905221765418 p^{14} T^{7} + 123594633620 p^{21} T^{8} + 372398 p^{28} T^{9} + p^{35} T^{10} $
	31	$C_2 \wr S_5$	$ 1 - 271372 T + 112879086507 T^{2} - 19060065197093520 T^{3} + $$49\!\cdots\!98$$ T^{4} - $$64\!\cdots\!92$$ T^{5} + $$49\!\cdots\!98$$ p^{7} T^{6} - 19060065197093520 p^{14} T^{7} + 112879086507 p^{21} T^{8} - 271372 p^{28} T^{9} + p^{35} T^{10} $
	37	$C_2 \wr S_5$	$ 1 + 562630 T + 503027979609 T^{2} + 5044147202230440 p T^{3} + $$94\!\cdots\!90$$ T^{4} + $$25\!\cdots\!60$$ T^{5} + $$94\!\cdots\!90$$ p^{7} T^{6} + 5044147202230440 p^{15} T^{7} + 503027979609 p^{21} T^{8} + 562630 p^{28} T^{9} + p^{35} T^{10} $
	41	$C_2 \wr S_5$	$ 1 + 956714 T + 858829677401 T^{2} + 558274454732676000 T^{3} + $$30\!\cdots\!54$$ T^{4} + $$14\!\cdots\!52$$ T^{5} + $$30\!\cdots\!54$$ p^{7} T^{6} + 558274454732676000 p^{14} T^{7} + 858829677401 p^{21} T^{8} + 956714 p^{28} T^{9} + p^{35} T^{10} $
	43	$C_2 \wr S_5$	$ 1 - 827362 T + 1446200884344 T^{2} - 842028213283761024 T^{3} + $$80\!\cdots\!43$$ T^{4} - $$33\!\cdots\!88$$ T^{5} + $$80\!\cdots\!43$$ p^{7} T^{6} - 842028213283761024 p^{14} T^{7} + 1446200884344 p^{21} T^{8} - 827362 p^{28} T^{9} + p^{35} T^{10} $
	47	$C_2 \wr S_5$	$ 1 - 1812982 T + 2671332480764 T^{2} - 2520094723128764352 T^{3} + $$23\!\cdots\!95$$ T^{4} - $$16\!\cdots\!84$$ T^{5} + $$23\!\cdots\!95$$ p^{7} T^{6} - 2520094723128764352 p^{14} T^{7} + 2671332480764 p^{21} T^{8} - 1812982 p^{28} T^{9} + p^{35} T^{10} $
	53	$C_2 \wr S_5$	$ 1 - 486998 T + 4629026089784 T^{2} - 1938463029671914986 T^{3} + $$98\!\cdots\!83$$ T^{4} - $$32\!\cdots\!92$$ T^{5} + $$98\!\cdots\!83$$ p^{7} T^{6} - 1938463029671914986 p^{14} T^{7} + 4629026089784 p^{21} T^{8} - 486998 p^{28} T^{9} + p^{35} T^{10} $
	59	$C_2 \wr S_5$	$ 1 - 367182 T - 1643557954748 T^{2} + 6627210193123102656 T^{3} + $$41\!\cdots\!31$$ T^{4} - $$14\!\cdots\!44$$ T^{5} + $$41\!\cdots\!31$$ p^{7} T^{6} + 6627210193123102656 p^{14} T^{7} - 1643557954748 p^{21} T^{8} - 367182 p^{28} T^{9} + p^{35} T^{10} $
	61	$C_2 \wr S_5$	$ 1 - 1879732 T + 7749404395290 T^{2} - 9963704865691638198 T^{3} + $$38\!\cdots\!65$$ T^{4} - $$47\!\cdots\!92$$ T^{5} + $$38\!\cdots\!65$$ p^{7} T^{6} - 9963704865691638198 p^{14} T^{7} + 7749404395290 p^{21} T^{8} - 1879732 p^{28} T^{9} + p^{35} T^{10} $
	67	$C_2 \wr S_5$	$ 1 - 1046394 T + 10097473571576 T^{2} - 25105239457925557932 T^{3} + $$64\!\cdots\!63$$ T^{4} - $$25\!\cdots\!36$$ T^{5} + $$64\!\cdots\!63$$ p^{7} T^{6} - 25105239457925557932 p^{14} T^{7} + 10097473571576 p^{21} T^{8} - 1046394 p^{28} T^{9} + p^{35} T^{10} $
	71	$C_2 \wr S_5$	$ 1 - 4664572 T + 12023820502163 T^{2} - 22827464116687888320 T^{3} + $$99\!\cdots\!02$$ T^{4} - $$47\!\cdots\!84$$ T^{5} + $$99\!\cdots\!02$$ p^{7} T^{6} - 22827464116687888320 p^{14} T^{7} + 12023820502163 p^{21} T^{8} - 4664572 p^{28} T^{9} + p^{35} T^{10} $
	73	$C_2 \wr S_5$	$ 1 - 4224942 T + 41896684548731 T^{2} - $$13\!\cdots\!28$$ T^{3} + $$81\!\cdots\!65$$ T^{4} - $$20\!\cdots\!14$$ T^{5} + $$81\!\cdots\!65$$ p^{7} T^{6} - $$13\!\cdots\!28$$ p^{14} T^{7} + 41896684548731 p^{21} T^{8} - 4224942 p^{28} T^{9} + p^{35} T^{10} $
	79	$C_2 \wr S_5$	$ 1 + 9574024 T + 1156078839417 p T^{2} + $$53\!\cdots\!00$$ T^{3} + $$29\!\cdots\!38$$ T^{4} + $$13\!\cdots\!04$$ T^{5} + $$29\!\cdots\!38$$ p^{7} T^{6} + $$53\!\cdots\!00$$ p^{14} T^{7} + 1156078839417 p^{22} T^{8} + 9574024 p^{28} T^{9} + p^{35} T^{10} $
	83	$C_2 \wr S_5$	$ 1 + 11754804 T + 110268615554371 T^{2} + $$80\!\cdots\!76$$ T^{3} + $$51\!\cdots\!90$$ T^{4} + $$27\!\cdots\!08$$ T^{5} + $$51\!\cdots\!90$$ p^{7} T^{6} + $$80\!\cdots\!76$$ p^{14} T^{7} + 110268615554371 p^{21} T^{8} + 11754804 p^{28} T^{9} + p^{35} T^{10} $
	89	$C_2 \wr S_5$	$ 1 - 2782542 T + 110940525514693 T^{2} - $$23\!\cdots\!64$$ T^{3} + $$78\!\cdots\!14$$ T^{4} - $$16\!\cdots\!36$$ T^{5} + $$78\!\cdots\!14$$ p^{7} T^{6} - $$23\!\cdots\!64$$ p^{14} T^{7} + 110940525514693 p^{21} T^{8} - 2782542 p^{28} T^{9} + p^{35} T^{10} $
	97	$C_2 \wr S_5$	$ 1 - 1291574 T + 106423285884609 T^{2} + $$22\!\cdots\!16$$ T^{3} + $$91\!\cdots\!50$$ T^{4} + $$61\!\cdots\!12$$ T^{5} + $$91\!\cdots\!50$$ p^{7} T^{6} + $$22\!\cdots\!16$$ p^{14} T^{7} + 106423285884609 p^{21} T^{8} - 1291574 p^{28} T^{9} + p^{35} T^{10} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−5.76208162582917275665198039881, −5.68324159604097040226170169716, −5.57585974775920311922556688918, −5.45730669705339249633925586178, −5.35447556242835158267082135221, −4.82596928105120134856593682261, −4.57550349868171073813748120721, −4.42275778742300385261739256571, −4.11499797330727975125547068589, −4.00950348388086529139939416153, −3.77242884245232527441699459975, −3.31796424516467783051429875016, −3.21467954785191994429783577483, −3.14955073063317548210295242922, −3.14007306604022485109688800318, −2.39901136922814952911494347670, −2.23995864856812135571020778222, −2.09498836223953812762325290198, −1.86495267848180990464092802082, −1.62807867140860183615997976104, −1.12007105681502797905691612403, −0.933800812564880253087117381702, −0.39002947168823220052555654092, −0.35876191394717689347071871160, −0.31862935928461645295582652365, 0.31862935928461645295582652365, 0.35876191394717689347071871160, 0.39002947168823220052555654092, 0.933800812564880253087117381702, 1.12007105681502797905691612403, 1.62807867140860183615997976104, 1.86495267848180990464092802082, 2.09498836223953812762325290198, 2.23995864856812135571020778222, 2.39901136922814952911494347670, 3.14007306604022485109688800318, 3.14955073063317548210295242922, 3.21467954785191994429783577483, 3.31796424516467783051429875016, 3.77242884245232527441699459975, 4.00950348388086529139939416153, 4.11499797330727975125547068589, 4.42275778742300385261739256571, 4.57550349868171073813748120721, 4.82596928105120134856593682261, 5.35447556242835158267082135221, 5.45730669705339249633925586178, 5.57585974775920311922556688918, 5.68324159604097040226170169716, 5.76208162582917275665198039881

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(2.702338824\)
\(L(\frac12)\)	\(\approx\)	\(2.702338824\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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