LMFDB - L-function 2-3332-17.16-c1-0-50

Dirichlet series

L(s) = 1

− 1.41i·3^-s − 2.82i·5^-s + 0.999·9^-s − 1.41i·11^-s + 4·13^-s − 4.00·15^-s + (−3 + 2.82i)17^-s + 4·19^-s − 1.41i·23^-s − 3.00·25^-s − 5.65i·27^-s + 2.82i·29^-s − 4.24i·31^-s − 2.00·33^-s − 8.48i·37^-s + ⋯

L(s) = 1

− 0.816i·3^-s − 1.26i·5^-s + 0.333·9^-s − 0.426i·11^-s + 1.10·13^-s − 1.03·15^-s + (−0.727 + 0.685i)17^-s + 0.917·19^-s − 0.294i·23^-s − 0.600·25^-s − 1.08i·27^-s + 0.525i·29^-s − 0.762i·31^-s − 0.348·33^-s − 1.39i·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3332$ = $2^{2} \cdot 7^{2} \cdot 17$
Sign:	$-0.727 + 0.685i$
Analytic conductor:	$26.6061$
Root analytic conductor:	$5.15811$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3332} (2549, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3332,\ (\ :1/2),\ -0.727 + 0.685i)$

Particular Values

$L(1)$	$\approx$	$2.018405627$
$L(\frac12)$	$\approx$	$2.018405627$
$L(\frac{3}{2})$		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 $
	7	$ 1 $
	17	$ 1 + (3 - 2.82i)T $
good	3	$ 1 + 1.41iT - 3T^{2} $
	5	$ 1 + 2.82iT - 5T^{2} $
	11	$ 1 + 1.41iT - 11T^{2} $
	13	$ 1 - 4T + 13T^{2} $
	19	$ 1 - 4T + 19T^{2} $
	23	$ 1 + 1.41iT - 23T^{2} $
	29	$ 1 - 2.82iT - 29T^{2} $
	31	$ 1 + 4.24iT - 31T^{2} $
	37	$ 1 + 8.48iT - 37T^{2} $
	41	$ 1 + 11.3iT - 41T^{2} $
	43	$ 1 - 8T + 43T^{2} $
	47	$ 1 - 12T + 47T^{2} $
	53	$ 1 + 6T + 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 8.48iT - 61T^{2} $
	67	$ 1 + 4T + 67T^{2} $
	71	$ 1 - 7.07iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 - 4.24iT - 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 + 12T + 89T^{2} $
	97	$ 1 - 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

−8.478853418615852924070730515768, −7.58244010693781146472357456071, −6.98330175918415174496554911145, −5.92416336827612259671591408493, −5.55971352873620900131879208340, −4.33556329033103752053931528617, −3.84723019786387171980301979499, −2.40401879973578705823224118485, −1.39188626362306616951778318912, −0.69197666760147965336248626507, 1.37450596698300350519268658808, 2.73350639811331207734201520511, 3.35557915310758406773997657842, 4.23651806592263843561796399330, 4.94533604186819072799102090721, 5.99073947578360479849892432441, 6.71585799606762800398881691780, 7.30460009731033303507926265645, 8.117150631464355883057345796486, 9.171924702527574337555717665426

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.018405627\)
\(L(\frac12)\)	\(\approx\)	\(2.018405627\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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