LMFDB - L-function 2-2340-13.12-c1-0-10

Dirichlet series

L(s) = 1

− i·5^-s − 2.76i·7^-s + 2.16i·11^-s + (−0.167 + 3.60i)13^-s + 5.03i·19^-s + 4.93·23^-s − 25^-s + 4.43·29^-s − 3.37i·31^-s − 2.76·35^-s + 3.97i·37^-s + 1.66i·41^-s + 9.80·43^-s + 11.6i·47^-s − 0.665·49^-s + ⋯

L(s) = 1

− 0.447i·5^-s − 1.04i·7^-s + 0.653i·11^-s + (−0.0463 + 0.998i)13^-s + 1.15i·19^-s + 1.02·23^-s − 0.200·25^-s + 0.823·29^-s − 0.605i·31^-s − 0.468·35^-s + 0.653i·37^-s + 0.260i·41^-s + 1.49·43^-s + 1.69i·47^-s − 0.0951·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2340$ = $2^{2} \cdot 3^{2} \cdot 5 \cdot 13$
Sign:	$0.998 + 0.0463i$
Analytic conductor:	$18.6849$
Root analytic conductor:	$4.32261$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2340} (181, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2340,\ (\ :1/2),\ 0.998 + 0.0463i)$

Particular Values

$L(1)$	$\approx$	$1.783866162$
$L(\frac12)$	$\approx$	$1.783866162$
$L(\frac{3}{2})$		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 $
	5	$ 1 + iT $
	13	$ 1 + (0.167 - 3.60i)T $
good	7	$ 1 + 2.76iT - 7T^{2} $
	11	$ 1 - 2.16iT - 11T^{2} $
	17	$ 1 + 17T^{2} $
	19	$ 1 - 5.03iT - 19T^{2} $
	23	$ 1 - 4.93T + 23T^{2} $
	29	$ 1 - 4.43T + 29T^{2} $
	31	$ 1 + 3.37iT - 31T^{2} $
	37	$ 1 - 3.97iT - 37T^{2} $
	41	$ 1 - 1.66iT - 41T^{2} $
	43	$ 1 - 9.80T + 43T^{2} $
	47	$ 1 - 11.6iT - 47T^{2} $
	53	$ 1 - 12.7T + 53T^{2} $
	59	$ 1 + 8.16iT - 59T^{2} $
	61	$ 1 + 10.3T + 61T^{2} $
	67	$ 1 + 7.10iT - 67T^{2} $
	71	$ 1 + 16.1iT - 71T^{2} $
	73	$ 1 + 15.9iT - 73T^{2} $
	79	$ 1 - 11.9T + 79T^{2} $
	83	$ 1 - 3.97iT - 83T^{2} $
	89	$ 1 + 7.94iT - 89T^{2} $
	97	$ 1 - 0.462iT - 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

−9.176729463716614888138972594897, −8.068027724007890703152461139574, −7.50752275492634223665112221328, −6.71802740336617988140173459727, −5.94807882913201412714265774558, −4.71414261574394912987728733569, −4.34210460240264325268139481714, −3.33049680801616701805898349897, −1.98572615896784423909858631651, −0.952613530724991696520135973400, 0.813781680327905000814484323365, 2.52274854458130399872027337840, 2.92222282986635740207571752738, 4.09743838947792036830666820661, 5.36759018431193465145157145202, 5.63108910034262353480548789549, 6.75745138819917264158408087677, 7.33937414515041251213328730309, 8.529364664912121467049746495069, 8.751800327361507420080530800000

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.783866162\)
\(L(\frac12)\)	\(\approx\)	\(1.783866162\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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