LMFDB - L-function 10-13e5-1.1-c9e5-0-0

Dirichlet series

L(s) = 1

+ 15·2^-s + 161·3^-s − 987·4^-s + 1.80e3·5^-s + 2.41e3·6^-s + 1.00e4·7^-s − 1.07e4·8^-s − 5.71e3·9^-s + 2.70e4·10^-s + 1.21e5·11^-s − 1.58e5·12^-s + 1.42e5·13^-s + 1.51e5·14^-s + 2.90e5·15^-s + 4.33e5·16^-s − 4.95e5·17^-s − 8.57e4·18^-s − 8.40e5·19^-s − 1.77e6·20^-s + 1.62e6·21^-s + 1.82e6·22^-s − 5.92e5·23^-s − 1.73e6·24^-s − 2.42e6·25^-s + 2.14e6·26^-s − 1.08e6·27^-s − 9.96e6·28^-s + ⋯

L(s) = 1

+ 0.662·2^-s + 1.14·3^-s − 1.92·4^-s + 1.29·5^-s + 0.760·6^-s + 1.58·7^-s − 0.929·8^-s − 0.290·9^-s + 0.855·10^-s + 2.50·11^-s − 2.21·12^-s + 1.38·13^-s + 1.05·14^-s + 1.48·15^-s + 1.65·16^-s − 1.43·17^-s − 0.192·18^-s − 1.48·19^-s − 2.48·20^-s + 1.82·21^-s + 1.66·22^-s − 0.441·23^-s − 1.06·24^-s − 1.24·25^-s + 0.919·26^-s − 0.392·27^-s − 3.06·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 371293 ^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 371293 ^{s/2} \, \Gamma_{\C}(s+9/2)^{5} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$10$
Conductor:	$371293$ = $13^{5}$
Sign:	$1$
Analytic conductor:	$13455.6$
Root analytic conductor:	$2.58755$
Motivic weight:	$9$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(10,\ 371293,\ (\ :9/2, 9/2, 9/2, 9/2, 9/2),\ 1)$

Particular Values

$L(5)$	$\approx$	$9.322764616$
$L(\frac12)$	$\approx$	$9.322764616$
$L(\frac{11}{2})$		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	13	$C_1$	$ ( 1 - p^{4} T )^{5} $
good	2	$C_2 \wr S_5$	$ 1 - 15 T + 303 p^{2} T^{2} - 5553 p^{2} T^{3} + 29201 p^{5} T^{4} - 252033 p^{6} T^{5} + 29201 p^{14} T^{6} - 5553 p^{20} T^{7} + 303 p^{29} T^{8} - 15 p^{36} T^{9} + p^{45} T^{10} $
	3	$C_2 \wr S_5$	$ 1 - 161 T + 10546 p T^{2} - 60851 p^{4} T^{3} + 37844987 p^{3} T^{4} - 1621483792 p^{4} T^{5} + 37844987 p^{12} T^{6} - 60851 p^{22} T^{7} + 10546 p^{28} T^{8} - 161 p^{36} T^{9} + p^{45} T^{10} $
	5	$C_2 \wr S_5$	$ 1 - 1803 T + 5673036 T^{2} - 2122118037 p T^{3} + 154127491547 p^{3} T^{4} - 217433194071432 p^{3} T^{5} + 154127491547 p^{12} T^{6} - 2122118037 p^{19} T^{7} + 5673036 p^{27} T^{8} - 1803 p^{36} T^{9} + p^{45} T^{10} $
	7	$C_2 \wr S_5$	$ 1 - 10099 T + 139254174 T^{2} - 122453891247 p T^{3} + 166521050648013 p^{2} T^{4} - 122374113130618344 p^{3} T^{5} + 166521050648013 p^{11} T^{6} - 122453891247 p^{19} T^{7} + 139254174 p^{27} T^{8} - 10099 p^{36} T^{9} + p^{45} T^{10} $
	11	$C_2 \wr S_5$	$ 1 - 121746 T + 14269408587 T^{2} - 1047597531024144 T^{3} + 70977489220411684558 T^{4} - $$36\!\cdots\!52$$ T^{5} + 70977489220411684558 p^{9} T^{6} - 1047597531024144 p^{18} T^{7} + 14269408587 p^{27} T^{8} - 121746 p^{36} T^{9} + p^{45} T^{10} $
	17	$C_2 \wr S_5$	$ 1 + 29157 p T + 303748065600 T^{2} + 136238774144131095 T^{3} + $$32\!\cdots\!83$$ p T^{4} + $$62\!\cdots\!64$$ p^{2} T^{5} + $$32\!\cdots\!83$$ p^{10} T^{6} + 136238774144131095 p^{18} T^{7} + 303748065600 p^{27} T^{8} + 29157 p^{37} T^{9} + p^{45} T^{10} $
	19	$C_2 \wr S_5$	$ 1 + 840738 T + 1352453971283 T^{2} + 953145467206471696 T^{3} + $$83\!\cdots\!34$$ T^{4} + $$43\!\cdots\!52$$ T^{5} + $$83\!\cdots\!34$$ p^{9} T^{6} + 953145467206471696 p^{18} T^{7} + 1352453971283 p^{27} T^{8} + 840738 p^{36} T^{9} + p^{45} T^{10} $
	23	$C_2 \wr S_5$	$ 1 + 592152 T + 5512736304883 T^{2} - 937525593298362720 T^{3} + $$10\!\cdots\!94$$ T^{4} - $$33\!\cdots\!76$$ p T^{5} + $$10\!\cdots\!94$$ p^{9} T^{6} - 937525593298362720 p^{18} T^{7} + 5512736304883 p^{27} T^{8} + 592152 p^{36} T^{9} + p^{45} T^{10} $
	29	$C_2 \wr S_5$	$ 1 - 10678182 T + 88796663366937 T^{2} - $$54\!\cdots\!56$$ T^{3} + $$28\!\cdots\!78$$ T^{4} - $$11\!\cdots\!44$$ T^{5} + $$28\!\cdots\!78$$ p^{9} T^{6} - $$54\!\cdots\!56$$ p^{18} T^{7} + 88796663366937 p^{27} T^{8} - 10678182 p^{36} T^{9} + p^{45} T^{10} $
	31	$C_2 \wr S_5$	$ 1 - 12885296 T + 164536395807867 T^{2} - $$12\!\cdots\!24$$ T^{3} + $$92\!\cdots\!78$$ T^{4} - $$48\!\cdots\!72$$ T^{5} + $$92\!\cdots\!78$$ p^{9} T^{6} - $$12\!\cdots\!24$$ p^{18} T^{7} + 164536395807867 p^{27} T^{8} - 12885296 p^{36} T^{9} + p^{45} T^{10} $
	37	$C_2 \wr S_5$	$ 1 - 7171823 T + 354892872282468 T^{2} - $$10\!\cdots\!13$$ T^{3} + $$60\!\cdots\!79$$ T^{4} - $$91\!\cdots\!16$$ T^{5} + $$60\!\cdots\!79$$ p^{9} T^{6} - $$10\!\cdots\!13$$ p^{18} T^{7} + 354892872282468 p^{27} T^{8} - 7171823 p^{36} T^{9} + p^{45} T^{10} $
	41	$C_2 \wr S_5$	$ 1 - 9294012 T + 571994210722321 T^{2} - $$89\!\cdots\!60$$ T^{3} + $$16\!\cdots\!02$$ T^{4} - $$34\!\cdots\!04$$ T^{5} + $$16\!\cdots\!02$$ p^{9} T^{6} - $$89\!\cdots\!60$$ p^{18} T^{7} + 571994210722321 p^{27} T^{8} - 9294012 p^{36} T^{9} + p^{45} T^{10} $
	43	$C_2 \wr S_5$	$ 1 - 12831975 T + 322694905923638 T^{2} - $$26\!\cdots\!89$$ T^{3} + $$62\!\cdots\!77$$ T^{4} - $$43\!\cdots\!96$$ T^{5} + $$62\!\cdots\!77$$ p^{9} T^{6} - $$26\!\cdots\!89$$ p^{18} T^{7} + 322694905923638 p^{27} T^{8} - 12831975 p^{36} T^{9} + p^{45} T^{10} $
	47	$C_2 \wr S_5$	$ 1 - 43354215 T + 2527109677831630 T^{2} - $$75\!\cdots\!93$$ T^{3} + $$33\!\cdots\!41$$ T^{4} - $$71\!\cdots\!56$$ T^{5} + $$33\!\cdots\!41$$ p^{9} T^{6} - $$75\!\cdots\!93$$ p^{18} T^{7} + 2527109677831630 p^{27} T^{8} - 43354215 p^{36} T^{9} + p^{45} T^{10} $
	53	$C_2 \wr S_5$	$ 1 - 93231780 T + 14353270896409645 T^{2} - $$11\!\cdots\!96$$ T^{3} + $$88\!\cdots\!06$$ T^{4} - $$52\!\cdots\!08$$ T^{5} + $$88\!\cdots\!06$$ p^{9} T^{6} - $$11\!\cdots\!96$$ p^{18} T^{7} + 14353270896409645 p^{27} T^{8} - 93231780 p^{36} T^{9} + p^{45} T^{10} $
	59	$C_2 \wr S_5$	$ 1 - 246496182 T + 33223507046890507 T^{2} - $$26\!\cdots\!60$$ T^{3} + $$17\!\cdots\!50$$ T^{4} - $$12\!\cdots\!36$$ T^{5} + $$17\!\cdots\!50$$ p^{9} T^{6} - $$26\!\cdots\!60$$ p^{18} T^{7} + 33223507046890507 p^{27} T^{8} - 246496182 p^{36} T^{9} + p^{45} T^{10} $
	61	$C_2 \wr S_5$	$ 1 + 132232612 T + 51932556467342037 T^{2} + $$51\!\cdots\!12$$ T^{3} + $$11\!\cdots\!74$$ T^{4} + $$84\!\cdots\!48$$ T^{5} + $$11\!\cdots\!74$$ p^{9} T^{6} + $$51\!\cdots\!12$$ p^{18} T^{7} + 51932556467342037 p^{27} T^{8} + 132232612 p^{36} T^{9} + p^{45} T^{10} $
	67	$C_2 \wr S_5$	$ 1 + 369388534 T + 167732159711342355 T^{2} + $$38\!\cdots\!64$$ T^{3} + $$97\!\cdots\!58$$ T^{4} + $$15\!\cdots\!40$$ T^{5} + $$97\!\cdots\!58$$ p^{9} T^{6} + $$38\!\cdots\!64$$ p^{18} T^{7} + 167732159711342355 p^{27} T^{8} + 369388534 p^{36} T^{9} + p^{45} T^{10} $
	71	$C_2 \wr S_5$	$ 1 - 212150457 T + 126078700918544326 T^{2} - $$34\!\cdots\!35$$ T^{3} + $$83\!\cdots\!89$$ T^{4} - $$22\!\cdots\!08$$ T^{5} + $$83\!\cdots\!89$$ p^{9} T^{6} - $$34\!\cdots\!35$$ p^{18} T^{7} + 126078700918544326 p^{27} T^{8} - 212150457 p^{36} T^{9} + p^{45} T^{10} $
	73	$C_2 \wr S_5$	$ 1 + 252729806 T + 113432726759207253 T^{2} + $$15\!\cdots\!40$$ T^{3} + $$10\!\cdots\!58$$ T^{4} + $$17\!\cdots\!28$$ T^{5} + $$10\!\cdots\!58$$ p^{9} T^{6} + $$15\!\cdots\!40$$ p^{18} T^{7} + 113432726759207253 p^{27} T^{8} + 252729806 p^{36} T^{9} + p^{45} T^{10} $
	79	$C_2 \wr S_5$	$ 1 + 1247271728 T + 1135234985129373579 T^{2} + $$68\!\cdots\!96$$ T^{3} + $$33\!\cdots\!94$$ T^{4} + $$12\!\cdots\!92$$ T^{5} + $$33\!\cdots\!94$$ p^{9} T^{6} + $$68\!\cdots\!96$$ p^{18} T^{7} + 1135234985129373579 p^{27} T^{8} + 1247271728 p^{36} T^{9} + p^{45} T^{10} $
	83	$C_2 \wr S_5$	$ 1 - 1696894296 T + 1900005505469354847 T^{2} - $$14\!\cdots\!20$$ T^{3} + $$90\!\cdots\!82$$ T^{4} - $$43\!\cdots\!16$$ T^{5} + $$90\!\cdots\!82$$ p^{9} T^{6} - $$14\!\cdots\!20$$ p^{18} T^{7} + 1900005505469354847 p^{27} T^{8} - 1696894296 p^{36} T^{9} + p^{45} T^{10} $
	89	$C_2 \wr S_5$	$ 1 + 753854382 T + 1391064645589335141 T^{2} + $$88\!\cdots\!08$$ T^{3} + $$90\!\cdots\!06$$ T^{4} + $$43\!\cdots\!00$$ T^{5} + $$90\!\cdots\!06$$ p^{9} T^{6} + $$88\!\cdots\!08$$ p^{18} T^{7} + 1391064645589335141 p^{27} T^{8} + 753854382 p^{36} T^{9} + p^{45} T^{10} $
	97	$C_2 \wr S_5$	$ 1 - 3824606 T + 1359157730156522205 T^{2} + $$14\!\cdots\!24$$ T^{3} + $$16\!\cdots\!14$$ T^{4} + $$77\!\cdots\!52$$ T^{5} + $$16\!\cdots\!14$$ p^{9} T^{6} + $$14\!\cdots\!24$$ p^{18} T^{7} + 1359157730156522205 p^{27} T^{8} - 3824606 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

−10.96755344068652021440322682103, −10.47146690063545857380741999627, −10.30120333071670590147466338689, −9.689771883234677143779068138526, −9.640976171901390060997929183005, −8.955033076950461752009246256058, −8.807407888513218374358739284116, −8.602688657565501795948227062440, −8.521660255554272723527879648770, −8.172037666415081195687127974513, −7.63330555332640379495250506575, −6.77715371842856182111550555252, −6.43043112468585301663273465919, −5.99103095338263178456271172566, −5.98066825546955764352594321115, −4.94515656802875929033096965756, −4.61546529671658897304204988220, −4.40347901934242206136518659056, −3.92162465517986363555414252976, −3.89953980598280880035062853658, −2.53158953959929218221181292797, −2.48308230655511526929776622450, −1.41442314638227447995018807528, −1.36658821899086888741855790842, −0.55580831322115090404196232425, 0.55580831322115090404196232425, 1.36658821899086888741855790842, 1.41442314638227447995018807528, 2.48308230655511526929776622450, 2.53158953959929218221181292797, 3.89953980598280880035062853658, 3.92162465517986363555414252976, 4.40347901934242206136518659056, 4.61546529671658897304204988220, 4.94515656802875929033096965756, 5.98066825546955764352594321115, 5.99103095338263178456271172566, 6.43043112468585301663273465919, 6.77715371842856182111550555252, 7.63330555332640379495250506575, 8.172037666415081195687127974513, 8.521660255554272723527879648770, 8.602688657565501795948227062440, 8.807407888513218374358739284116, 8.955033076950461752009246256058, 9.640976171901390060997929183005, 9.689771883234677143779068138526, 10.30120333071670590147466338689, 10.47146690063545857380741999627, 10.96755344068652021440322682103

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(9.322764616\)
\(L(\frac12)\)	\(\approx\)	\(9.322764616\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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