LMFDB - L-function 10-546e5-1.1-c7e5-0-0

Dirichlet series

L(s) = 1

− 40·2^-s + 135·3^-s + 960·4^-s + 509·5^-s − 5.40e3·6^-s − 1.71e3·7^-s − 1.79e4·8^-s + 1.09e4·9^-s − 2.03e4·10^-s + 958·11^-s + 1.29e5·12^-s + 1.09e4·13^-s + 6.86e4·14^-s + 6.87e4·15^-s + 2.86e5·16^-s + 7.86e3·17^-s − 4.37e5·18^-s + 6.01e4·19^-s + 4.88e5·20^-s − 2.31e5·21^-s − 3.83e4·22^-s + 1.22e5·23^-s − 2.41e6·24^-s − 2.89e4·25^-s − 4.39e5·26^-s + 6.88e5·27^-s − 1.64e6·28^-s + ⋯

L(s) = 1

− 3.53·2^-s + 2.88·3^-s + 15/2·4^-s + 1.82·5^-s − 10.2·6^-s − 1.88·7^-s − 12.3·8^-s + 5·9^-s − 6.43·10^-s + 0.217·11^-s + 21.6·12^-s + 1.38·13^-s + 6.68·14^-s + 5.25·15^-s + 35/2·16^-s + 0.388·17^-s − 17.6·18^-s + 2.01·19^-s + 13.6·20^-s − 5.45·21^-s − 0.767·22^-s + 2.10·23^-s − 35.7·24^-s − 0.370·25^-s − 4.90·26^-s + 6.73·27^-s − 14.1·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$10$
Conductor:	$2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5}$
Sign:	$1$
Analytic conductor:	$1.44349\times 10^{11}$
Root analytic conductor:	$13.0599$
Motivic weight:	$7$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(10,\ 2^{5} \cdot 3^{5} \cdot 7^{5} \cdot 13^{5} ,\ ( \ : 7/2, 7/2, 7/2, 7/2, 7/2 ),\ 1 )$

Particular Values

$L(4)$	$\approx$	$24.79913011$
$L(\frac12)$	$\approx$	$24.79913011$
$L(\frac{9}{2})$		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	$C_1$	$ ( 1 + p^{3} T )^{5} $
	3	$C_1$	$ ( 1 - p^{3} T )^{5} $
	7	$C_1$	$ ( 1 + p^{3} T )^{5} $
	13	$C_1$	$ ( 1 - p^{3} T )^{5} $
good	5	$C_2 \wr S_5$	$ 1 - 509 T + 287992 T^{2} - 97836451 T^{3} + 6888447231 p T^{4} - 386920863136 p^{2} T^{5} + 6888447231 p^{8} T^{6} - 97836451 p^{14} T^{7} + 287992 p^{21} T^{8} - 509 p^{28} T^{9} + p^{35} T^{10} $
	11	$C_2 \wr S_5$	$ 1 - 958 T + 46514310 T^{2} - 3125802102 p T^{3} + 1227776035144841 T^{4} - 2103878293043376 p^{2} T^{5} + 1227776035144841 p^{7} T^{6} - 3125802102 p^{15} T^{7} + 46514310 p^{21} T^{8} - 958 p^{28} T^{9} + p^{35} T^{10} $
	17	$C_2 \wr S_5$	$ 1 - 7864 T + 1334922512 T^{2} - 5416571140296 T^{3} + 51052793162792935 p T^{4} - $$23\!\cdots\!72$$ T^{5} + 51052793162792935 p^{8} T^{6} - 5416571140296 p^{14} T^{7} + 1334922512 p^{21} T^{8} - 7864 p^{28} T^{9} + p^{35} T^{10} $
	19	$C_2 \wr S_5$	$ 1 - 3167 p T + 4144797478 T^{2} - 179034345968535 T^{3} + 7158028100298253441 T^{4} - $$22\!\cdots\!68$$ T^{5} + 7158028100298253441 p^{7} T^{6} - 179034345968535 p^{14} T^{7} + 4144797478 p^{21} T^{8} - 3167 p^{29} T^{9} + p^{35} T^{10} $
	23	$C_2 \wr S_5$	$ 1 - 122869 T + 20771335958 T^{2} - 1636818330993203 T^{3} + $$15\!\cdots\!01$$ T^{4} - $$82\!\cdots\!24$$ T^{5} + $$15\!\cdots\!01$$ p^{7} T^{6} - 1636818330993203 p^{14} T^{7} + 20771335958 p^{21} T^{8} - 122869 p^{28} T^{9} + p^{35} T^{10} $
	29	$C_2 \wr S_5$	$ 1 + 17317 T + 50798747632 T^{2} + 2584999675225587 T^{3} + $$11\!\cdots\!23$$ T^{4} + $$76\!\cdots\!36$$ T^{5} + $$11\!\cdots\!23$$ p^{7} T^{6} + 2584999675225587 p^{14} T^{7} + 50798747632 p^{21} T^{8} + 17317 p^{28} T^{9} + p^{35} T^{10} $
	31	$C_2 \wr S_5$	$ 1 + 177665 T + 59968078655 T^{2} + 9001111418133340 T^{3} + $$21\!\cdots\!66$$ T^{4} + $$20\!\cdots\!42$$ T^{5} + $$21\!\cdots\!66$$ p^{7} T^{6} + 9001111418133340 p^{14} T^{7} + 59968078655 p^{21} T^{8} + 177665 p^{28} T^{9} + p^{35} T^{10} $
	37	$C_2 \wr S_5$	$ 1 + 55136 T + 303778060792 T^{2} + 5363332033782364 T^{3} + $$46\!\cdots\!15$$ T^{4} + $$51\!\cdots\!48$$ T^{5} + $$46\!\cdots\!15$$ p^{7} T^{6} + 5363332033782364 p^{14} T^{7} + 303778060792 p^{21} T^{8} + 55136 p^{28} T^{9} + p^{35} T^{10} $
	41	$C_2 \wr S_5$	$ 1 + 237570 T + 515557432769 T^{2} + 196369940614048832 T^{3} + $$13\!\cdots\!10$$ T^{4} + $$59\!\cdots\!60$$ T^{5} + $$13\!\cdots\!10$$ p^{7} T^{6} + 196369940614048832 p^{14} T^{7} + 515557432769 p^{21} T^{8} + 237570 p^{28} T^{9} + p^{35} T^{10} $
	43	$C_2 \wr S_5$	$ 1 + 970601 T + 1195111865590 T^{2} + 702743687833631347 T^{3} + $$50\!\cdots\!65$$ T^{4} + $$23\!\cdots\!60$$ T^{5} + $$50\!\cdots\!65$$ p^{7} T^{6} + 702743687833631347 p^{14} T^{7} + 1195111865590 p^{21} T^{8} + 970601 p^{28} T^{9} + p^{35} T^{10} $
	47	$C_2 \wr S_5$	$ 1 + 384035 T + 1529042617435 T^{2} + 617198546939589940 T^{3} + $$11\!\cdots\!26$$ T^{4} + $$43\!\cdots\!22$$ T^{5} + $$11\!\cdots\!26$$ p^{7} T^{6} + 617198546939589940 p^{14} T^{7} + 1529042617435 p^{21} T^{8} + 384035 p^{28} T^{9} + p^{35} T^{10} $
	53	$C_2 \wr S_5$	$ 1 + 1977 T + 2819181392741 T^{2} - 456715899235884572 T^{3} + $$44\!\cdots\!02$$ T^{4} - $$79\!\cdots\!02$$ T^{5} + $$44\!\cdots\!02$$ p^{7} T^{6} - 456715899235884572 p^{14} T^{7} + 2819181392741 p^{21} T^{8} + 1977 p^{28} T^{9} + p^{35} T^{10} $
	59	$C_2 \wr S_5$	$ 1 - 2057936 T + 4864887855599 T^{2} - 4907900100221024832 T^{3} + $$15\!\cdots\!62$$ T^{4} - $$19\!\cdots\!72$$ T^{5} + $$15\!\cdots\!62$$ p^{7} T^{6} - 4907900100221024832 p^{14} T^{7} + 4864887855599 p^{21} T^{8} - 2057936 p^{28} T^{9} + p^{35} T^{10} $
	61	$C_2 \wr S_5$	$ 1 + 723756 T + 8116610328538 T^{2} + 10578325797076970314 T^{3} + $$37\!\cdots\!09$$ T^{4} + $$46\!\cdots\!84$$ T^{5} + $$37\!\cdots\!09$$ p^{7} T^{6} + 10578325797076970314 p^{14} T^{7} + 8116610328538 p^{21} T^{8} + 723756 p^{28} T^{9} + p^{35} T^{10} $
	67	$C_2 \wr S_5$	$ 1 - 2695018 T + 2784967158803 T^{2} + 3956964041696313872 T^{3} - $$15\!\cdots\!62$$ T^{4} - $$12\!\cdots\!00$$ T^{5} - $$15\!\cdots\!62$$ p^{7} T^{6} + 3956964041696313872 p^{14} T^{7} + 2784967158803 p^{21} T^{8} - 2695018 p^{28} T^{9} + p^{35} T^{10} $
	71	$C_2 \wr S_5$	$ 1 - 7392916 T + 61658982486035 T^{2} - $$27\!\cdots\!72$$ T^{3} + $$12\!\cdots\!26$$ T^{4} - $$37\!\cdots\!72$$ T^{5} + $$12\!\cdots\!26$$ p^{7} T^{6} - $$27\!\cdots\!72$$ p^{14} T^{7} + 61658982486035 p^{21} T^{8} - 7392916 p^{28} T^{9} + p^{35} T^{10} $
	73	$C_2 \wr S_5$	$ 1 - 8720441 T + 56998817220732 T^{2} - $$27\!\cdots\!23$$ T^{3} + $$11\!\cdots\!79$$ T^{4} - $$42\!\cdots\!00$$ T^{5} + $$11\!\cdots\!79$$ p^{7} T^{6} - $$27\!\cdots\!23$$ p^{14} T^{7} + 56998817220732 p^{21} T^{8} - 8720441 p^{28} T^{9} + p^{35} T^{10} $
	79	$C_2 \wr S_5$	$ 1 - 4646419 T + 79466497866523 T^{2} - $$36\!\cdots\!12$$ T^{3} + $$27\!\cdots\!06$$ T^{4} - $$10\!\cdots\!38$$ T^{5} + $$27\!\cdots\!06$$ p^{7} T^{6} - $$36\!\cdots\!12$$ p^{14} T^{7} + 79466497866523 p^{21} T^{8} - 4646419 p^{28} T^{9} + p^{35} T^{10} $
	83	$C_2 \wr S_5$	$ 1 - 17766733 T + 179482578636611 T^{2} - $$11\!\cdots\!36$$ T^{3} + $$59\!\cdots\!26$$ T^{4} - $$29\!\cdots\!62$$ T^{5} + $$59\!\cdots\!26$$ p^{7} T^{6} - $$11\!\cdots\!36$$ p^{14} T^{7} + 179482578636611 p^{21} T^{8} - 17766733 p^{28} T^{9} + p^{35} T^{10} $
	89	$C_2 \wr S_5$	$ 1 - 4692321 T + 137167117254481 T^{2} - $$10\!\cdots\!32$$ T^{3} + $$86\!\cdots\!78$$ T^{4} - $$71\!\cdots\!70$$ T^{5} + $$86\!\cdots\!78$$ p^{7} T^{6} - $$10\!\cdots\!32$$ p^{14} T^{7} + 137167117254481 p^{21} T^{8} - 4692321 p^{28} T^{9} + p^{35} T^{10} $
	97	$C_2 \wr S_5$	$ 1 - 15680305 T + 275561220183217 T^{2} - $$21\!\cdots\!80$$ T^{3} + $$22\!\cdots\!54$$ T^{4} - $$12\!\cdots\!26$$ T^{5} + $$22\!\cdots\!54$$ p^{7} T^{6} - $$21\!\cdots\!80$$ p^{14} T^{7} + 275561220183217 p^{21} T^{8} - 15680305 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

−5.33192936435813649399017882859, −5.26707762762996238223015351749, −5.25197414379302353944380673993, −5.05694166299398737688176677941, −4.87327000990895932168081053847, −3.79908386190290964149635938415, −3.77492868144901314025507558347, −3.70436329117543321015842921995, −3.70433652388673120088923363840, −3.68417640656809982959224892564, −3.08379247962324099626848097537, −2.77333417597186984436927697719, −2.76548726231080361080068847537, −2.73453201745408098689378228399, −2.60146306805393927797583587509, −1.90638696090712628355102476505, −1.76116268641482602730359084769, −1.75414583377411187577696936221, −1.74372157794805269828999231333, −1.61714450899840030626867943679, −0.948782190190546941993814076256, −0.76477542535796629619860296572, −0.72896784250869987560480705609, −0.56322852487641978707952873896, −0.40261386890436113168653379684, 0.40261386890436113168653379684, 0.56322852487641978707952873896, 0.72896784250869987560480705609, 0.76477542535796629619860296572, 0.948782190190546941993814076256, 1.61714450899840030626867943679, 1.74372157794805269828999231333, 1.75414583377411187577696936221, 1.76116268641482602730359084769, 1.90638696090712628355102476505, 2.60146306805393927797583587509, 2.73453201745408098689378228399, 2.76548726231080361080068847537, 2.77333417597186984436927697719, 3.08379247962324099626848097537, 3.68417640656809982959224892564, 3.70433652388673120088923363840, 3.70436329117543321015842921995, 3.77492868144901314025507558347, 3.79908386190290964149635938415, 4.87327000990895932168081053847, 5.05694166299398737688176677941, 5.25197414379302353944380673993, 5.26707762762996238223015351749, 5.33192936435813649399017882859

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(24.79913011\)
\(L(\frac12)\)	\(\approx\)	\(24.79913011\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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L-functions

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