LMFDB - L-function 10-315e5-1.1-c9e5-0-0

Dirichlet series

L(s) = 1

− 2·2^-s − 862·4^-s + 3.12e3·5^-s + 1.20e4·7^-s + 6.78e3·8^-s − 6.25e3·10^-s − 1.03e4·11^-s + 1.58e5·13^-s − 2.40e4·14^-s + 1.23e5·16^-s − 3.16e4·17^-s + 1.65e6·19^-s − 2.69e6·20^-s + 2.06e4·22^-s − 7.96e5·23^-s + 5.85e6·25^-s − 3.17e5·26^-s − 1.03e7·28^-s + 3.35e6·29^-s + 2.67e6·31^-s − 6.04e6·32^-s + 6.32e4·34^-s + 3.75e7·35^-s + 5.09e7·37^-s − 3.31e6·38^-s + 2.11e7·40^-s − 6.33e6·41^-s + ⋯

L(s) = 1

− 0.0883·2^-s − 1.68·4^-s + 2.23·5^-s + 1.88·7^-s + 0.585·8^-s − 0.197·10^-s − 0.212·11^-s + 1.54·13^-s − 0.167·14^-s + 0.470·16^-s − 0.0918·17^-s + 2.91·19^-s − 3.76·20^-s + 0.0187·22^-s − 0.593·23^-s + 3·25^-s − 0.136·26^-s − 3.18·28^-s + 0.880·29^-s + 0.520·31^-s − 1.01·32^-s + 0.00811·34^-s + 4.22·35^-s + 4.47·37^-s − 0.257·38^-s + 1.30·40^-s − 0.349·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{5} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 5^{5} \cdot 7^{5}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree:	$10$
Conductor:	$3^{10} \cdot 5^{5} \cdot 7^{5}$
Sign:	$1$
Analytic conductor:	$1.12393\times 10^{11}$
Root analytic conductor:	$12.7372$
Motivic weight:	$9$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(10,\ 3^{10} \cdot 5^{5} \cdot 7^{5} ,\ ( \ : 9/2, 9/2, 9/2, 9/2, 9/2 ),\ 1 )$

Particular Values

$L(5)$	$\approx$	$75.53948046$
$L(\frac12)$	$\approx$	$75.53948046$
$L(\frac{11}{2})$		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	3		$ 1 $
	5	$C_1$	$ ( 1 - p^{4} T )^{5} $
	7	$C_1$	$ ( 1 - p^{4} T )^{5} $
good	2	$C_2 \wr S_5$	$ 1 + p T + 433 p T^{2} - 831 p^{2} T^{3} + 18845 p^{5} T^{4} - 27043 p^{6} T^{5} + 18845 p^{14} T^{6} - 831 p^{20} T^{7} + 433 p^{28} T^{8} + p^{37} T^{9} + p^{45} T^{10} $
	11	$C_2 \wr S_5$	$ 1 + 10312 T + 6110501469 T^{2} - 12868286470036 T^{3} + 18721446338695361377 T^{4} - $$14\!\cdots\!88$$ T^{5} + 18721446338695361377 p^{9} T^{6} - 12868286470036 p^{18} T^{7} + 6110501469 p^{27} T^{8} + 10312 p^{36} T^{9} + p^{45} T^{10} $
	13	$C_2 \wr S_5$	$ 1 - 158638 T + 40099411663 T^{2} - 4525202776113992 T^{3} + $$74\!\cdots\!09$$ T^{4} - $$64\!\cdots\!38$$ T^{5} + $$74\!\cdots\!09$$ p^{9} T^{6} - 4525202776113992 p^{18} T^{7} + 40099411663 p^{27} T^{8} - 158638 p^{36} T^{9} + p^{45} T^{10} $
	17	$C_2 \wr S_5$	$ 1 + 31614 T + 152979949219 T^{2} + 15467440208130592 T^{3} + $$26\!\cdots\!33$$ T^{4} + $$20\!\cdots\!30$$ T^{5} + $$26\!\cdots\!33$$ p^{9} T^{6} + 15467440208130592 p^{18} T^{7} + 152979949219 p^{27} T^{8} + 31614 p^{36} T^{9} + p^{45} T^{10} $
	19	$C_2 \wr S_5$	$ 1 - 1655376 T + 1555849159655 T^{2} - 44845180156018720 p T^{3} + $$28\!\cdots\!70$$ T^{4} - $$95\!\cdots\!08$$ T^{5} + $$28\!\cdots\!70$$ p^{9} T^{6} - 44845180156018720 p^{19} T^{7} + 1555849159655 p^{27} T^{8} - 1655376 p^{36} T^{9} + p^{45} T^{10} $
	23	$C_2 \wr S_5$	$ 1 + 796104 T + 3980003153371 T^{2} + 4279841785977132736 T^{3} + $$51\!\cdots\!98$$ p T^{4} + $$93\!\cdots\!40$$ T^{5} + $$51\!\cdots\!98$$ p^{10} T^{6} + 4279841785977132736 p^{18} T^{7} + 3980003153371 p^{27} T^{8} + 796104 p^{36} T^{9} + p^{45} T^{10} $
	29	$C_2 \wr S_5$	$ 1 - 3353726 T + 28116887828895 T^{2} - $$11\!\cdots\!00$$ T^{3} + $$68\!\cdots\!45$$ T^{4} - $$18\!\cdots\!78$$ T^{5} + $$68\!\cdots\!45$$ p^{9} T^{6} - $$11\!\cdots\!00$$ p^{18} T^{7} + 28116887828895 p^{27} T^{8} - 3353726 p^{36} T^{9} + p^{45} T^{10} $
	31	$C_2 \wr S_5$	$ 1 - 2678120 T + 30980429107403 T^{2} - $$17\!\cdots\!84$$ T^{3} + $$13\!\cdots\!94$$ T^{4} - $$47\!\cdots\!88$$ T^{5} + $$13\!\cdots\!94$$ p^{9} T^{6} - $$17\!\cdots\!84$$ p^{18} T^{7} + 30980429107403 p^{27} T^{8} - 2678120 p^{36} T^{9} + p^{45} T^{10} $
	37	$C_2 \wr S_5$	$ 1 - 50994846 T + 39149870417693 p T^{2} - $$27\!\cdots\!92$$ T^{3} + $$40\!\cdots\!66$$ T^{4} - $$49\!\cdots\!08$$ T^{5} + $$40\!\cdots\!66$$ p^{9} T^{6} - $$27\!\cdots\!92$$ p^{18} T^{7} + 39149870417693 p^{28} T^{8} - 50994846 p^{36} T^{9} + p^{45} T^{10} $
	41	$C_2 \wr S_5$	$ 1 + 6330194 T + 1562261358649949 T^{2} + $$78\!\cdots\!48$$ T^{3} + $$99\!\cdots\!30$$ T^{4} + $$38\!\cdots\!76$$ T^{5} + $$99\!\cdots\!30$$ p^{9} T^{6} + $$78\!\cdots\!48$$ p^{18} T^{7} + 1562261358649949 p^{27} T^{8} + 6330194 p^{36} T^{9} + p^{45} T^{10} $
	43	$C_2 \wr S_5$	$ 1 - 6149468 T + 2230226649262079 T^{2} - $$11\!\cdots\!80$$ T^{3} + $$21\!\cdots\!34$$ T^{4} - $$88\!\cdots\!36$$ T^{5} + $$21\!\cdots\!34$$ p^{9} T^{6} - $$11\!\cdots\!80$$ p^{18} T^{7} + 2230226649262079 p^{27} T^{8} - 6149468 p^{36} T^{9} + p^{45} T^{10} $
	47	$C_2 \wr S_5$	$ 1 - 6897780 T + 2892726247790561 T^{2} + $$13\!\cdots\!96$$ T^{3} + $$42\!\cdots\!25$$ T^{4} + $$40\!\cdots\!32$$ T^{5} + $$42\!\cdots\!25$$ p^{9} T^{6} + $$13\!\cdots\!96$$ p^{18} T^{7} + 2892726247790561 p^{27} T^{8} - 6897780 p^{36} T^{9} + p^{45} T^{10} $
	53	$C_2 \wr S_5$	$ 1 - 70886738 T + 8460080414958121 T^{2} + $$37\!\cdots\!40$$ T^{3} - $$39\!\cdots\!94$$ T^{4} + $$21\!\cdots\!32$$ T^{5} - $$39\!\cdots\!94$$ p^{9} T^{6} + $$37\!\cdots\!40$$ p^{18} T^{7} + 8460080414958121 p^{27} T^{8} - 70886738 p^{36} T^{9} + p^{45} T^{10} $
	59	$C_2 \wr S_5$	$ 1 - 2581952 p T + 34448863513048391 T^{2} - $$43\!\cdots\!68$$ T^{3} + $$52\!\cdots\!42$$ T^{4} - $$51\!\cdots\!48$$ T^{5} + $$52\!\cdots\!42$$ p^{9} T^{6} - $$43\!\cdots\!68$$ p^{18} T^{7} + 34448863513048391 p^{27} T^{8} - 2581952 p^{37} T^{9} + p^{45} T^{10} $
	61	$C_2 \wr S_5$	$ 1 - 257015698 T + 23075149340036769 T^{2} - $$27\!\cdots\!72$$ T^{3} - $$13\!\cdots\!50$$ T^{4} + $$20\!\cdots\!60$$ T^{5} - $$13\!\cdots\!50$$ p^{9} T^{6} - $$27\!\cdots\!72$$ p^{18} T^{7} + 23075149340036769 p^{27} T^{8} - 257015698 p^{36} T^{9} + p^{45} T^{10} $
	67	$C_2 \wr S_5$	$ 1 - 133467828 T + 56930250312806863 T^{2} + $$14\!\cdots\!40$$ T^{3} + $$54\!\cdots\!62$$ T^{4} + $$24\!\cdots\!00$$ T^{5} + $$54\!\cdots\!62$$ p^{9} T^{6} + $$14\!\cdots\!40$$ p^{18} T^{7} + 56930250312806863 p^{27} T^{8} - 133467828 p^{36} T^{9} + p^{45} T^{10} $
	71	$C_2 \wr S_5$	$ 1 + 522788960 T + 303569112441738147 T^{2} + $$93\!\cdots\!00$$ T^{3} + $$30\!\cdots\!82$$ T^{4} + $$63\!\cdots\!80$$ T^{5} + $$30\!\cdots\!82$$ p^{9} T^{6} + $$93\!\cdots\!00$$ p^{18} T^{7} + 303569112441738147 p^{27} T^{8} + 522788960 p^{36} T^{9} + p^{45} T^{10} $
	73	$C_2 \wr S_5$	$ 1 - 159370858 T + 184580562002358805 T^{2} - $$29\!\cdots\!44$$ T^{3} + $$17\!\cdots\!22$$ T^{4} - $$23\!\cdots\!16$$ T^{5} + $$17\!\cdots\!22$$ p^{9} T^{6} - $$29\!\cdots\!44$$ p^{18} T^{7} + 184580562002358805 p^{27} T^{8} - 159370858 p^{36} T^{9} + p^{45} T^{10} $
	79	$C_2 \wr S_5$	$ 1 - 464174900 T + 596055521003027425 T^{2} - $$19\!\cdots\!64$$ T^{3} + $$13\!\cdots\!85$$ T^{4} - $$33\!\cdots\!32$$ T^{5} + $$13\!\cdots\!85$$ p^{9} T^{6} - $$19\!\cdots\!64$$ p^{18} T^{7} + 596055521003027425 p^{27} T^{8} - 464174900 p^{36} T^{9} + p^{45} T^{10} $
	83	$C_2 \wr S_5$	$ 1 + 2207636832 T + 2812258639953749663 T^{2} + $$24\!\cdots\!80$$ T^{3} + $$15\!\cdots\!98$$ T^{4} + $$76\!\cdots\!76$$ T^{5} + $$15\!\cdots\!98$$ p^{9} T^{6} + $$24\!\cdots\!80$$ p^{18} T^{7} + 2812258639953749663 p^{27} T^{8} + 2207636832 p^{36} T^{9} + p^{45} T^{10} $
	89	$C_2 \wr S_5$	$ 1 - 1106708326 T + 1689084835940499725 T^{2} - $$13\!\cdots\!80$$ T^{3} + $$11\!\cdots\!90$$ T^{4} - $$68\!\cdots\!28$$ T^{5} + $$11\!\cdots\!90$$ p^{9} T^{6} - $$13\!\cdots\!80$$ p^{18} T^{7} + 1689084835940499725 p^{27} T^{8} - 1106708326 p^{36} T^{9} + p^{45} T^{10} $
	97	$C_2 \wr S_5$	$ 1 - 1956142254 T + 4082161263087973027 T^{2} - $$52\!\cdots\!20$$ T^{3} + $$63\!\cdots\!73$$ T^{4} - $$57\!\cdots\!62$$ T^{5} + $$63\!\cdots\!73$$ p^{9} T^{6} - $$52\!\cdots\!20$$ p^{18} T^{7} + 4082161263087973027 p^{27} T^{8} - 1956142254 p^{36} T^{9} + p^{45} 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

−5.41383426369081855983460714443, −5.32428738747807592715270538774, −5.18568863678018082211440493726, −5.16121767966419257553177566298, −5.05241587647620470837437701755, −4.38930595338330187375075741352, −4.25567832939604624265446161751, −4.18771062113447505601598990517, −4.14716068245433914139797409119, −4.09174836166329894029723432041, −3.33970869044726039218822183672, −3.09292665285183073700157081443, −3.06990573541549945708675248060, −2.72871894271178897455531388988, −2.49946131448995770865052251661, −2.15971062799013188676711471893, −2.13816663465035840005448159418, −1.61870746603028050587192357780, −1.56814324479734969900044241340, −1.43211930527215041113664991390, −1.06237923557917822673925662931, −0.73894454603894110129504733716, −0.63894109696943917112414608145, −0.62783666133123237192773690870, −0.58331875452740775976452465584, 0.58331875452740775976452465584, 0.62783666133123237192773690870, 0.63894109696943917112414608145, 0.73894454603894110129504733716, 1.06237923557917822673925662931, 1.43211930527215041113664991390, 1.56814324479734969900044241340, 1.61870746603028050587192357780, 2.13816663465035840005448159418, 2.15971062799013188676711471893, 2.49946131448995770865052251661, 2.72871894271178897455531388988, 3.06990573541549945708675248060, 3.09292665285183073700157081443, 3.33970869044726039218822183672, 4.09174836166329894029723432041, 4.14716068245433914139797409119, 4.18771062113447505601598990517, 4.25567832939604624265446161751, 4.38930595338330187375075741352, 5.05241587647620470837437701755, 5.16121767966419257553177566298, 5.18568863678018082211440493726, 5.32428738747807592715270538774, 5.41383426369081855983460714443

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(75.53948046\)
\(L(\frac12)\)	\(\approx\)	\(75.53948046\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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