LMFDB - L-function 2-4032-21.20-c1-0-4

Dirichlet series

L(s) = 1

− 3.69·5^-s + (−2.41 + 1.08i)7^-s − 3.41i·11^-s − 5.22i·13^-s − 6.75·17^-s − 2.16i·19^-s + 6.24i·23^-s + 8.65·25^-s − 2.58i·29^-s − 10.4i·31^-s + (8.92 − 4i)35^-s − 4·37^-s + 0.634·41^-s − 6.48·43^-s − 3.06·47^-s + ⋯

L(s) = 1

− 1.65·5^-s + (−0.912 + 0.409i)7^-s − 1.02i·11^-s − 1.44i·13^-s − 1.63·17^-s − 0.496i·19^-s + 1.30i·23^-s + 1.73·25^-s − 0.480i·29^-s − 1.87i·31^-s + (1.50 − 0.676i)35^-s − 0.657·37^-s + 0.0990·41^-s − 0.988·43^-s − 0.446·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.192 - 0.981i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (3905, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ 0.192 - 0.981i)$

Particular Values

$L(1)$	$\approx$	$0.1808635190$
$L(\frac12)$	$\approx$	$0.1808635190$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	7	$ 1 + (2.41 - 1.08i)T $
good	5	$ 1 + 3.69T + 5T^{2} $
	11	$ 1 + 3.41iT - 11T^{2} $
	13	$ 1 + 5.22iT - 13T^{2} $
	17	$ 1 + 6.75T + 17T^{2} $
	19	$ 1 + 2.16iT - 19T^{2} $
	23	$ 1 - 6.24iT - 23T^{2} $
	29	$ 1 + 2.58iT - 29T^{2} $
	31	$ 1 + 10.4iT - 31T^{2} $
	37	$ 1 + 4T + 37T^{2} $
	41	$ 1 - 0.634T + 41T^{2} $
	43	$ 1 + 6.48T + 43T^{2} $
	47	$ 1 + 3.06T + 47T^{2} $
	53	$ 1 - 2.58iT - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 1.65T + 67T^{2} $
	71	$ 1 + 7.41iT - 71T^{2} $
	73	$ 1 + 0.896iT - 73T^{2} $
	79	$ 1 - 13.6T + 79T^{2} $
	83	$ 1 + 11.7T + 83T^{2} $
	89	$ 1 + 6.75T + 89T^{2} $
	97	$ 1 - 9.55iT - 97T^{2} $
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$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.481784233460382781125997853657, −7.931745629643975844845765523977, −7.26003148836723317435789974920, −6.41509726789770436280586765066, −5.70428537278397112234620731490, −4.78142939377948906830361400785, −3.78113931355172037089949364792, −3.34567528949381492016485147120, −2.47932400773668284890602011379, −0.58816187871477779800986081694, 0.093068359409457587351585262317, 1.71940199412349086901178426883, 2.90817570446132317028905865822, 3.85464330999064793977237409191, 4.34795099386091908907337166468, 4.93642372007060374213463876380, 6.61329291547778845564823642078, 6.77970180083890831708651099448, 7.32333014051764916953801243409, 8.463402186637613615161068244090

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1808635190\)
\(L(\frac12)\)	\(\approx\)	\(0.1808635190\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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