LMFDB - L-function 2-2320-29.28-c1-0-26

Dirichlet series

L(s) = 1

− 1.41i·3^-s − 5^-s + 2.73·7^-s + 0.999·9^-s − 0.378i·11^-s + 5.46·13^-s + 1.41i·15^-s + 3.48i·17^-s + 4.24i·19^-s − 3.86i·21^-s − 2.19·23^-s + 25^-s − 5.65i·27^-s + (5.19 + 1.41i)29^-s + 4.24i·31^-s + ⋯

L(s) = 1

− 0.816i·3^-s − 0.447·5^-s + 1.03·7^-s + 0.333·9^-s − 0.114i·11^-s + 1.51·13^-s + 0.365i·15^-s + 0.845i·17^-s + 0.973i·19^-s − 0.843i·21^-s − 0.457·23^-s + 0.200·25^-s − 1.08i·27^-s + (0.964 + 0.262i)29^-s + 0.762i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2320$ = $2^{4} \cdot 5 \cdot 29$
Sign:	$0.964 + 0.262i$
Analytic conductor:	$18.5252$
Root analytic conductor:	$4.30410$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2320} (1681, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2320,\ (\ :1/2),\ 0.964 + 0.262i)$

Particular Values

$L(1)$	$\approx$	$2.140465880$
$L(\frac12)$	$\approx$	$2.140465880$
$L(\frac{3}{2})$		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 $
	5	$ 1 + T $
	29	$ 1 + (-5.19 - 1.41i)T $
good	3	$ 1 + 1.41iT - 3T^{2} $
	7	$ 1 - 2.73T + 7T^{2} $
	11	$ 1 + 0.378iT - 11T^{2} $
	13	$ 1 - 5.46T + 13T^{2} $
	17	$ 1 - 3.48iT - 17T^{2} $
	19	$ 1 - 4.24iT - 19T^{2} $
	23	$ 1 + 2.19T + 23T^{2} $
	31	$ 1 - 4.24iT - 31T^{2} $
	37	$ 1 - 4.24iT - 37T^{2} $
	41	$ 1 - 5.93iT - 41T^{2} $
	43	$ 1 - 4.24iT - 43T^{2} $
	47	$ 1 - 11.2iT - 47T^{2} $
	53	$ 1 + 53T^{2} $
	59	$ 1 + 6T + 59T^{2} $
	61	$ 1 + 11.5iT - 61T^{2} $
	67	$ 1 - 13.1T + 67T^{2} $
	71	$ 1 + 6T + 71T^{2} $
	73	$ 1 + 15.8iT - 73T^{2} $
	79	$ 1 + 1.13iT - 79T^{2} $
	83	$ 1 - 8.19T + 83T^{2} $
	89	$ 1 - 7.72iT - 89T^{2} $
	97	$ 1 - 7.34iT - 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

−8.565831099601683828653970150659, −8.087375977248019769396706986620, −7.71473948602675444592189910973, −6.46360284952031574797069357621, −6.18529818981322819974045782285, −4.90955088820831404944897835855, −4.13162452186157690293185474822, −3.21001717308991377220075045661, −1.69938768256210903696223887381, −1.21745934827391173854219011712, 0.907199724828692192729413359244, 2.23263135057711408546083913699, 3.54712748094132065713288754443, 4.21395927081618065266961046696, 4.89365679511686793205808391214, 5.69633403724829802711995551895, 6.83907882159094368529177608496, 7.51108649733405599930184536904, 8.479970438063116086206922405722, 8.875993741515632467892366075239

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.140465880\)
\(L(\frac12)\)	\(\approx\)	\(2.140465880\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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