LMFDB - L-function 8-15e4-1.1-c27e4-0-1

Dirichlet series

L(s) = 1

− 1.97e4·2^-s + 6.37e6·3^-s + 1.22e8·4^-s − 4.88e9·5^-s − 1.25e11·6^-s + 3.51e11·7^-s − 3.55e11·8^-s + 2.54e13·9^-s + 9.63e13·10^-s − 2.89e13·11^-s + 7.78e14·12^-s − 5.69e14·13^-s − 6.94e15·14^-s − 3.11e16·15^-s − 3.76e15·16^-s − 7.15e15·17^-s − 5.01e17·18^-s − 3.37e16·19^-s − 5.96e17·20^-s + 2.24e18·21^-s + 5.71e17·22^-s − 8.61e18·23^-s − 2.26e18·24^-s + 1.49e19·25^-s + 1.12e19·26^-s + 8.10e19·27^-s + 4.29e19·28^-s + ⋯

L(s) = 1

− 1.70·2^-s + 2.30·3^-s + 0.909·4^-s − 1.78·5^-s − 3.93·6^-s + 1.37·7^-s − 0.228·8^-s + 10/3·9^-s + 3.04·10^-s − 0.253·11^-s + 2.10·12^-s − 0.521·13^-s − 2.33·14^-s − 4.13·15^-s − 0.208·16^-s − 0.175·17^-s − 5.67·18^-s − 0.184·19^-s − 1.62·20^-s + 3.17·21^-s + 0.431·22^-s − 3.56·23^-s − 0.527·24^-s + 2·25^-s + 0.887·26^-s + 3.84·27^-s + 1.24·28^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+27/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$50625$ = $3^{4} \cdot 5^{4}$
Sign:	$1$
Analytic conductor:	$2.30350\times 10^{7}$
Root analytic conductor:	$8.32336$
Motivic weight:	$27$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$4$
Selberg data:	$(8,\ 50625,\ (\ :27/2, 27/2, 27/2, 27/2),\ 1)$

Particular Values

$L(14)$	$=$	$0$
$L(\frac12)$	$=$	$0$
$L(\frac{29}{2})$		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	$C_1$	$ ( 1 - p^{13} T )^{4} $
	5	$C_1$	$ ( 1 + p^{13} T )^{4} $
good	2	$C_2 \wr S_4$	$ 1 + 9867 p T + 16707455 p^{4} T^{2} + 6290879177 p^{9} T^{3} + 318094206003 p^{17} T^{4} + 6290879177 p^{36} T^{5} + 16707455 p^{58} T^{6} + 9867 p^{82} T^{7} + p^{108} T^{8} $
	7	$C_2 \wr S_4$	$ 1 - 7182531488 p^{2} T + $$32\!\cdots\!32$$ p^{2} T^{2} - $$12\!\cdots\!44$$ p^{5} T^{3} + $$32\!\cdots\!50$$ p^{4} T^{4} - $$12\!\cdots\!44$$ p^{32} T^{5} + $$32\!\cdots\!32$$ p^{56} T^{6} - 7182531488 p^{83} T^{7} + p^{108} T^{8} $
	11	$C_2 \wr S_4$	$ 1 + 28973681140224 T + $$22\!\cdots\!72$$ p T^{2} - $$89\!\cdots\!20$$ p^{2} T^{3} + $$19\!\cdots\!66$$ p^{4} T^{4} - $$89\!\cdots\!20$$ p^{29} T^{5} + $$22\!\cdots\!72$$ p^{55} T^{6} + 28973681140224 p^{81} T^{7} + p^{108} T^{8} $
	13	$C_2 \wr S_4$	$ 1 + 43775418691048 p T + $$75\!\cdots\!84$$ p^{2} T^{2} + $$90\!\cdots\!56$$ p^{3} T^{3} + $$77\!\cdots\!50$$ p^{4} T^{4} + $$90\!\cdots\!56$$ p^{30} T^{5} + $$75\!\cdots\!84$$ p^{56} T^{6} + 43775418691048 p^{82} T^{7} + p^{108} T^{8} $
	17	$C_2 \wr S_4$	$ 1 + 7152765494513496 T + $$31\!\cdots\!44$$ p T^{2} + $$96\!\cdots\!80$$ p^{2} T^{3} + $$25\!\cdots\!02$$ p^{3} T^{4} + $$96\!\cdots\!80$$ p^{29} T^{5} + $$31\!\cdots\!44$$ p^{55} T^{6} + 7152765494513496 p^{81} T^{7} + p^{108} T^{8} $
	19	$C_2 \wr S_4$	$ 1 + 33753130743907168 T + $$47\!\cdots\!88$$ p T^{2} - $$13\!\cdots\!64$$ p^{2} T^{3} + $$54\!\cdots\!86$$ p^{3} T^{4} - $$13\!\cdots\!64$$ p^{29} T^{5} + $$47\!\cdots\!88$$ p^{55} T^{6} + 33753130743907168 p^{81} T^{7} + p^{108} T^{8} $
	23	$C_2 \wr S_4$	$ 1 + 374557423878711792 p T + $$92\!\cdots\!08$$ p^{2} T^{2} + $$14\!\cdots\!88$$ p^{3} T^{3} + $$18\!\cdots\!90$$ p^{4} T^{4} + $$14\!\cdots\!88$$ p^{30} T^{5} + $$92\!\cdots\!08$$ p^{56} T^{6} + 374557423878711792 p^{82} T^{7} + p^{108} T^{8} $
	29	$C_2 \wr S_4$	$ 1 + 1665772391398631736 T - $$10\!\cdots\!20$$ T^{2} - $$63\!\cdots\!68$$ T^{3} + $$15\!\cdots\!78$$ T^{4} - $$63\!\cdots\!68$$ p^{27} T^{5} - $$10\!\cdots\!20$$ p^{54} T^{6} + 1665772391398631736 p^{81} T^{7} + p^{108} T^{8} $
	31	$C_2 \wr S_4$	$ 1 + 73781948023144578736 T + $$54\!\cdots\!68$$ T^{2} + $$22\!\cdots\!88$$ T^{3} + $$12\!\cdots\!54$$ T^{4} + $$22\!\cdots\!88$$ p^{27} T^{5} + $$54\!\cdots\!68$$ p^{54} T^{6} + 73781948023144578736 p^{81} T^{7} + p^{108} T^{8} $
	37	$C_2 \wr S_4$	$ 1 - $$86\!\cdots\!32$$ T + $$42\!\cdots\!28$$ T^{2} - $$13\!\cdots\!08$$ T^{3} + $$88\!\cdots\!70$$ T^{4} - $$13\!\cdots\!08$$ p^{27} T^{5} + $$42\!\cdots\!28$$ p^{54} T^{6} - $$86\!\cdots\!32$$ p^{81} T^{7} + p^{108} T^{8} $
	41	$C_2 \wr S_4$	$ 1 + $$12\!\cdots\!76$$ T + $$27\!\cdots\!68$$ p T^{2} + $$58\!\cdots\!28$$ T^{3} + $$35\!\cdots\!34$$ T^{4} + $$58\!\cdots\!28$$ p^{27} T^{5} + $$27\!\cdots\!68$$ p^{55} T^{6} + $$12\!\cdots\!76$$ p^{81} T^{7} + p^{108} T^{8} $
	43	$C_2 \wr S_4$	$ 1 - $$24\!\cdots\!40$$ T + $$61\!\cdots\!20$$ T^{2} - $$88\!\cdots\!80$$ T^{3} + $$12\!\cdots\!98$$ T^{4} - $$88\!\cdots\!80$$ p^{27} T^{5} + $$61\!\cdots\!20$$ p^{54} T^{6} - $$24\!\cdots\!40$$ p^{81} T^{7} + p^{108} T^{8} $
	47	$C_2 \wr S_4$	$ 1 + $$12\!\cdots\!40$$ T + $$46\!\cdots\!80$$ T^{2} + $$63\!\cdots\!20$$ T^{3} + $$91\!\cdots\!38$$ T^{4} + $$63\!\cdots\!20$$ p^{27} T^{5} + $$46\!\cdots\!80$$ p^{54} T^{6} + $$12\!\cdots\!40$$ p^{81} T^{7} + p^{108} T^{8} $
	53	$C_2 \wr S_4$	$ 1 + $$73\!\cdots\!16$$ T + $$30\!\cdots\!12$$ T^{2} + $$87\!\cdots\!76$$ T^{3} + $$18\!\cdots\!50$$ T^{4} + $$87\!\cdots\!76$$ p^{27} T^{5} + $$30\!\cdots\!12$$ p^{54} T^{6} + $$73\!\cdots\!16$$ p^{81} T^{7} + p^{108} T^{8} $
	59	$C_2 \wr S_4$	$ 1 - $$79\!\cdots\!88$$ T + $$24\!\cdots\!12$$ T^{2} - $$13\!\cdots\!16$$ T^{3} + $$22\!\cdots\!34$$ T^{4} - $$13\!\cdots\!16$$ p^{27} T^{5} + $$24\!\cdots\!12$$ p^{54} T^{6} - $$79\!\cdots\!88$$ p^{81} T^{7} + p^{108} T^{8} $
	61	$C_2 \wr S_4$	$ 1 - $$40\!\cdots\!40$$ T + $$39\!\cdots\!16$$ T^{2} - $$18\!\cdots\!60$$ T^{3} + $$82\!\cdots\!46$$ T^{4} - $$18\!\cdots\!60$$ p^{27} T^{5} + $$39\!\cdots\!16$$ p^{54} T^{6} - $$40\!\cdots\!40$$ p^{81} T^{7} + p^{108} T^{8} $
	67	$C_2 \wr S_4$	$ 1 - $$53\!\cdots\!64$$ T + $$65\!\cdots\!28$$ T^{2} - $$26\!\cdots\!00$$ T^{3} + $$17\!\cdots\!66$$ T^{4} - $$26\!\cdots\!00$$ p^{27} T^{5} + $$65\!\cdots\!28$$ p^{54} T^{6} - $$53\!\cdots\!64$$ p^{81} T^{7} + p^{108} T^{8} $
	71	$C_2 \wr S_4$	$ 1 + $$14\!\cdots\!12$$ T + $$25\!\cdots\!68$$ T^{2} + $$24\!\cdots\!84$$ T^{3} + $$30\!\cdots\!70$$ T^{4} + $$24\!\cdots\!84$$ p^{27} T^{5} + $$25\!\cdots\!68$$ p^{54} T^{6} + $$14\!\cdots\!12$$ p^{81} T^{7} + p^{108} T^{8} $
	73	$C_2 \wr S_4$	$ 1 + $$41\!\cdots\!36$$ T + $$13\!\cdots\!32$$ T^{2} + $$27\!\cdots\!36$$ T^{3} + $$45\!\cdots\!90$$ T^{4} + $$27\!\cdots\!36$$ p^{27} T^{5} + $$13\!\cdots\!32$$ p^{54} T^{6} + $$41\!\cdots\!36$$ p^{81} T^{7} + p^{108} T^{8} $
	79	$C_2 \wr S_4$	$ 1 - $$34\!\cdots\!80$$ T + $$37\!\cdots\!36$$ T^{2} - $$38\!\cdots\!60$$ T^{3} + $$90\!\cdots\!34$$ p T^{4} - $$38\!\cdots\!60$$ p^{27} T^{5} + $$37\!\cdots\!36$$ p^{54} T^{6} - $$34\!\cdots\!80$$ p^{81} T^{7} + p^{108} T^{8} $
	83	$C_2 \wr S_4$	$ 1 + $$32\!\cdots\!68$$ T + $$60\!\cdots\!40$$ T^{2} + $$74\!\cdots\!92$$ T^{3} + $$69\!\cdots\!46$$ T^{4} + $$74\!\cdots\!92$$ p^{27} T^{5} + $$60\!\cdots\!40$$ p^{54} T^{6} + $$32\!\cdots\!68$$ p^{81} T^{7} + p^{108} T^{8} $
	89	$C_2 \wr S_4$	$ 1 + $$26\!\cdots\!48$$ T + $$72\!\cdots\!52$$ T^{2} + $$11\!\cdots\!16$$ T^{3} + $$33\!\cdots\!34$$ T^{4} + $$11\!\cdots\!16$$ p^{27} T^{5} + $$72\!\cdots\!52$$ p^{54} T^{6} + $$26\!\cdots\!48$$ p^{81} T^{7} + p^{108} T^{8} $
	97	$C_2 \wr S_4$	$ 1 + $$26\!\cdots\!36$$ T + $$34\!\cdots\!88$$ T^{2} + $$30\!\cdots\!20$$ T^{3} + $$21\!\cdots\!86$$ T^{4} + $$30\!\cdots\!20$$ p^{27} T^{5} + $$34\!\cdots\!88$$ p^{54} T^{6} + $$26\!\cdots\!36$$ p^{81} T^{7} + p^{108} T^{8} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−9.645786507706348450426164510552, −8.785662663711480860844413638388, −8.587448474567917362637680191033, −8.508838293317693818572491279852, −8.354520875923876328974558589114, −7.81004849135266989118077360893, −7.68473987003525770043259842080, −7.64377270181993031363442793403, −7.19400155807380897794623107895, −6.60155518243812861761109347759, −6.37567162151094273721635410225, −5.57187450676840182897592198440, −5.28811418604824713914973154516, −4.52826924514269020663162641665, −4.29500672372175802411506517589, −4.16395844418772572917868743629, −4.06130851399824374063708228959, −3.20673148434097072229211243596, −3.15198191675583778470686801759, −2.76610443811211103260764408335, −2.18458475698317228836386692175, −2.09828437082055359723529559818, −1.43840774900393177627454773732, −1.43443024610090004240346298846, −1.12840139309614512216350554916, 0, 0, 0, 0, 1.12840139309614512216350554916, 1.43443024610090004240346298846, 1.43840774900393177627454773732, 2.09828437082055359723529559818, 2.18458475698317228836386692175, 2.76610443811211103260764408335, 3.15198191675583778470686801759, 3.20673148434097072229211243596, 4.06130851399824374063708228959, 4.16395844418772572917868743629, 4.29500672372175802411506517589, 4.52826924514269020663162641665, 5.28811418604824713914973154516, 5.57187450676840182897592198440, 6.37567162151094273721635410225, 6.60155518243812861761109347759, 7.19400155807380897794623107895, 7.64377270181993031363442793403, 7.68473987003525770043259842080, 7.81004849135266989118077360893, 8.354520875923876328974558589114, 8.508838293317693818572491279852, 8.587448474567917362637680191033, 8.785662663711480860844413638388, 9.645786507706348450426164510552

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 $\F_{q}$

\(L(14)\)	\(=\)	\(0\)
\(L(\frac12)\)	\(=\)	\(0\)
\(L(\frac{29}{2})\)		not available
\(L(1)\)		not available

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