LMFDB - L-function 8-5e8-1.1-c37e4-0-0

Dirichlet series

L(s) = 1

+ 2.37e11·4^-s + 1.79e18·9^-s − 5.34e19·11^-s + 9.95e21·16^-s − 7.46e23·19^-s + 2.54e27·29^-s + 5.22e26·31^-s + 4.26e29·36^-s − 2.52e30·41^-s − 1.26e31·44^-s + 4.09e31·49^-s + 4.75e32·59^-s + 2.03e32·61^-s − 4.13e33·64^-s − 1.49e33·71^-s − 1.77e35·76^-s − 5.44e35·79^-s + 2.02e36·81^-s + 2.66e36·89^-s − 9.62e37·99^-s − 3.66e35·101^-s − 5.89e37·109^-s + 6.02e38·116^-s + 7.30e38·121^-s + 1.23e38·124^-s + 127^-s + 131^-s + ⋯

L(s) = 1

+ 1.72·4^-s + 3.99·9^-s − 2.89·11^-s + 0.526·16^-s − 1.64·19^-s + 2.24·29^-s + 0.134·31^-s + 6.89·36^-s − 3.67·41^-s − 5.00·44^-s + 2.20·49^-s + 0.825·59^-s + 0.190·61^-s − 1.59·64^-s − 0.0842·71^-s − 2.83·76^-s − 4.26·79^-s + 9.97·81^-s + 2.30·89^-s − 11.5·99^-s − 0.0305·101^-s − 1.19·109^-s + 3.86·116^-s + 2.14·121^-s + 0.231·124^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 390625 ^{s/2} \, \Gamma_{\C}(s+37/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$8$
Conductor:	$390625$ = $5^{8}$
Sign:	$1$
Analytic conductor:	$2.20860\times 10^{9}$
Root analytic conductor:	$14.7236$
Motivic weight:	$37$
Rational:	yes
Arithmetic:	yes
Character:	Trivial
Primitive:	no
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(8,\ 390625,\ (\ :37/2, 37/2, 37/2, 37/2),\ 1)$

Particular Values

$L(19)$	$\approx$	$3.786308496$
$L(\frac12)$	$\approx$	$3.786308496$
$L(\frac{39}{2})$		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	5		$ 1 $
good	2	$D_4\times C_2$	$ 1 - 231599255 p^{10} T^{2} + 689777131079337 p^{26} T^{4} - 231599255 p^{84} T^{6} + p^{148} T^{8} $
	3	$D_4\times C_2$	$ 1 - 2468470768427660 p^{6} T^{2} + $$34\!\cdots\!38$$ p^{20} T^{4} - 2468470768427660 p^{80} T^{6} + p^{148} T^{8} $
	7	$D_4\times C_2$	$ 1 - $$17\!\cdots\!00$$ p^{4} T^{2} + $$33\!\cdots\!02$$ p^{10} T^{4} - $$17\!\cdots\!00$$ p^{78} T^{6} + p^{148} T^{8} $
	11	$D_{4}$	$ ( 1 + 2430366941349867096 p T + $$53\!\cdots\!46$$ p^{3} T^{2} + 2430366941349867096 p^{38} T^{3} + p^{74} T^{4} )^{2} $
	13	$D_4\times C_2$	$ 1 - $$21\!\cdots\!20$$ p^{2} T^{2} + $$13\!\cdots\!42$$ p^{6} T^{4} - $$21\!\cdots\!20$$ p^{76} T^{6} + p^{148} T^{8} $
	17	$D_4\times C_2$	$ 1 - $$32\!\cdots\!40$$ p^{2} T^{2} + $$17\!\cdots\!82$$ p^{6} T^{4} - $$32\!\cdots\!40$$ p^{76} T^{6} + p^{148} T^{8} $
	19	$D_{4}$	$ ( 1 + $$37\!\cdots\!20$$ T + $$10\!\cdots\!62$$ p T^{2} + $$37\!\cdots\!20$$ p^{37} T^{3} + p^{74} T^{4} )^{2} $
	23	$D_4\times C_2$	$ 1 - $$28\!\cdots\!20$$ T^{2} + $$38\!\cdots\!42$$ p^{2} T^{4} - $$28\!\cdots\!20$$ p^{74} T^{6} + p^{148} T^{8} $
	29	$D_{4}$	$ ( 1 - $$43\!\cdots\!80$$ p T + $$26\!\cdots\!98$$ p^{2} T^{2} - $$43\!\cdots\!80$$ p^{38} T^{3} + p^{74} T^{4} )^{2} $
	31	$D_{4}$	$ ( 1 - $$84\!\cdots\!04$$ p T + $$25\!\cdots\!06$$ p^{2} T^{2} - $$84\!\cdots\!04$$ p^{38} T^{3} + p^{74} T^{4} )^{2} $
	37	$D_4\times C_2$	$ 1 - $$21\!\cdots\!80$$ T^{2} + $$29\!\cdots\!78$$ T^{4} - $$21\!\cdots\!80$$ p^{74} T^{6} + p^{148} T^{8} $
	41	$D_{4}$	$ ( 1 + $$12\!\cdots\!36$$ T + $$12\!\cdots\!86$$ T^{2} + $$12\!\cdots\!36$$ p^{37} T^{3} + p^{74} T^{4} )^{2} $
	43	$D_4\times C_2$	$ 1 - $$22\!\cdots\!00$$ T^{2} - $$13\!\cdots\!02$$ T^{4} - $$22\!\cdots\!00$$ p^{74} T^{6} + p^{148} T^{8} $
	47	$D_4\times C_2$	$ 1 - $$28\!\cdots\!40$$ T^{2} + $$30\!\cdots\!38$$ T^{4} - $$28\!\cdots\!40$$ p^{74} T^{6} + p^{148} T^{8} $
	53	$D_4\times C_2$	$ 1 - $$84\!\cdots\!40$$ T^{2} + $$46\!\cdots\!38$$ T^{4} - $$84\!\cdots\!40$$ p^{74} T^{6} + p^{148} T^{8} $
	59	$D_{4}$	$ ( 1 - $$23\!\cdots\!40$$ T + $$64\!\cdots\!38$$ T^{2} - $$23\!\cdots\!40$$ p^{37} T^{3} + p^{74} T^{4} )^{2} $
	61	$D_{4}$	$ ( 1 - $$10\!\cdots\!44$$ T + $$10\!\cdots\!26$$ T^{2} - $$10\!\cdots\!44$$ p^{37} T^{3} + p^{74} T^{4} )^{2} $
	67	$D_4\times C_2$	$ 1 - $$69\!\cdots\!60$$ T^{2} + $$28\!\cdots\!58$$ T^{4} - $$69\!\cdots\!60$$ p^{74} T^{6} + p^{148} T^{8} $
	71	$D_{4}$	$ ( 1 + $$74\!\cdots\!16$$ T + $$58\!\cdots\!46$$ T^{2} + $$74\!\cdots\!16$$ p^{37} T^{3} + p^{74} T^{4} )^{2} $
	73	$D_4\times C_2$	$ 1 - $$33\!\cdots\!20$$ T^{2} + $$42\!\cdots\!18$$ T^{4} - $$33\!\cdots\!20$$ p^{74} T^{6} + p^{148} T^{8} $
	79	$D_{4}$	$ ( 1 + $$27\!\cdots\!80$$ T + $$64\!\cdots\!42$$ p T^{2} + $$27\!\cdots\!80$$ p^{37} T^{3} + p^{74} T^{4} )^{2} $
	83	$D_4\times C_2$	$ 1 - $$27\!\cdots\!60$$ T^{2} + $$38\!\cdots\!58$$ T^{4} - $$27\!\cdots\!60$$ p^{74} T^{6} + p^{148} T^{8} $
	89	$D_{4}$	$ ( 1 - $$13\!\cdots\!60$$ T + $$23\!\cdots\!58$$ T^{2} - $$13\!\cdots\!60$$ p^{37} T^{3} + p^{74} T^{4} )^{2} $
	97	$D_4\times C_2$	$ 1 - $$78\!\cdots\!40$$ T^{2} + $$30\!\cdots\!38$$ T^{4} - $$78\!\cdots\!40$$ p^{74} T^{6} + p^{148} 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

−7.16963935013912798873156370159, −6.86576296954759850702204769466, −6.60893528521541672257309134465, −6.41847882068679130062088845174, −6.20803092466880470575235927502, −5.55419398780132498861449836878, −5.36258687809282993913279347802, −4.99141963107884952760491815130, −4.65586780171470725972535602826, −4.65368637811279987078930293727, −4.27480837362406501285910272327, −3.89981183664887117745918828995, −3.82011241451543731876518533666, −3.19628568628580867001106655234, −2.90568439056820371127755326393, −2.65733701182515049661562570090, −2.46072723556122825635893930327, −2.15497441414689432035231039873, −1.75245722670016953137950360283, −1.74569867412075831408991212712, −1.57527619239360043289703522741, −1.12577152703225214822048256987, −0.65371497003403686505607208548, −0.64145346317658265134020937135, −0.12843252446085591193563613650, 0.12843252446085591193563613650, 0.64145346317658265134020937135, 0.65371497003403686505607208548, 1.12577152703225214822048256987, 1.57527619239360043289703522741, 1.74569867412075831408991212712, 1.75245722670016953137950360283, 2.15497441414689432035231039873, 2.46072723556122825635893930327, 2.65733701182515049661562570090, 2.90568439056820371127755326393, 3.19628568628580867001106655234, 3.82011241451543731876518533666, 3.89981183664887117745918828995, 4.27480837362406501285910272327, 4.65368637811279987078930293727, 4.65586780171470725972535602826, 4.99141963107884952760491815130, 5.36258687809282993913279347802, 5.55419398780132498861449836878, 6.20803092466880470575235927502, 6.41847882068679130062088845174, 6.60893528521541672257309134465, 6.86576296954759850702204769466, 7.16963935013912798873156370159

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 $\F_{q}$

\(L(19)\)	\(\approx\)	\(3.786308496\)
\(L(\frac12)\)	\(\approx\)	\(3.786308496\)
\(L(\frac{39}{2})\)		not available
\(L(1)\)		not available

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