LMFDB - L-function 2-3380-13.12-c1-0-1

Dirichlet series

L(s) = 1

− 2.33·3^-s + i·5^-s − 0.399i·7^-s + 2.43·9^-s + 1.73i·11^-s − 2.33i·15^-s − 0.692·17^-s + 5.37i·19^-s + 0.932i·21^-s + 0.107·23^-s − 25^-s + 1.30·27^-s − 4.90·29^-s − 7.86i·31^-s − 4.03i·33^-s + ⋯

L(s) = 1

− 1.34·3^-s + 0.447i·5^-s − 0.151i·7^-s + 0.813·9^-s + 0.522i·11^-s − 0.602i·15^-s − 0.167·17^-s + 1.23i·19^-s + 0.203i·21^-s + 0.0223·23^-s − 0.200·25^-s + 0.251·27^-s − 0.910·29^-s − 1.41i·31^-s − 0.703i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3380$ = $2^{2} \cdot 5 \cdot 13^{2}$
Sign:	$-0.960 + 0.277i$
Analytic conductor:	$26.9894$
Root analytic conductor:	$5.19513$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3380} (3041, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3380,\ (\ :1/2),\ -0.960 + 0.277i)$

Particular Values

$L(1)$	$\approx$	$0.1569378295$
$L(\frac12)$	$\approx$	$0.1569378295$
$L(\frac{3}{2})$		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 - iT $
	13	$ 1 $
good	3	$ 1 + 2.33T + 3T^{2} $
	7	$ 1 + 0.399iT - 7T^{2} $
	11	$ 1 - 1.73iT - 11T^{2} $
	17	$ 1 + 0.692T + 17T^{2} $
	19	$ 1 - 5.37iT - 19T^{2} $
	23	$ 1 - 0.107T + 23T^{2} $
	29	$ 1 + 4.90T + 29T^{2} $
	31	$ 1 + 7.86iT - 31T^{2} $
	37	$ 1 - 2.26iT - 37T^{2} $
	41	$ 1 - 7.73iT - 41T^{2} $
	43	$ 1 + 6.01T + 43T^{2} $
	47	$ 1 - 3.46iT - 47T^{2} $
	53	$ 1 - 11.7T + 53T^{2} $
	59	$ 1 - 7.27iT - 59T^{2} $
	61	$ 1 - 8.68T + 61T^{2} $
	67	$ 1 + 1.32iT - 67T^{2} $
	71	$ 1 - 3.87iT - 71T^{2} $
	73	$ 1 + 10.2iT - 73T^{2} $
	79	$ 1 + 13.1T + 79T^{2} $
	83	$ 1 + 14.0iT - 83T^{2} $
	89	$ 1 - 0.347iT - 89T^{2} $
	97	$ 1 - 8.85iT - 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

−9.132329286735085529658773844836, −8.086842057166257253861319962957, −7.38795018642855657237536555590, −6.65876916521293138133538392014, −5.95806669464593248887446927773, −5.43599164283332385510101056419, −4.47054060118645504378397257411, −3.75512473284014225096563328414, −2.49387704102364303022550732024, −1.32792801023467508076199930019, 0.06902947732918989446907537546, 1.06421093969370159514749891295, 2.39111294339031115176473140353, 3.62108154760282379684566341315, 4.56329113752725019771556759430, 5.45094787805754206614150220662, 5.59976656506954207708665208815, 6.82158184615124844798651271227, 7.06573784738294797522096058503, 8.414449443220147060720399257753

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.1569378295\)
\(L(\frac12)\)	\(\approx\)	\(0.1569378295\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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