LMFDB - L-function 2-3380-13.12-c1-0-35

Dirichlet series

L(s) = 1

− 3.40·3^-s − i·5^-s + 2.33i·7^-s + 8.60·9^-s − 6.16i·11^-s + 3.40i·15^-s − 3.07·17^-s + 0.0902i·19^-s − 7.96i·21^-s − 4.81·23^-s − 25^-s − 19.0·27^-s + 5.63·29^-s − 8.34i·31^-s + 21.0i·33^-s + ⋯

L(s) = 1

− 1.96·3^-s − 0.447i·5^-s + 0.883i·7^-s + 2.86·9^-s − 1.85i·11^-s + 0.879i·15^-s − 0.745·17^-s + 0.0207i·19^-s − 1.73i·21^-s − 1.00·23^-s − 0.200·25^-s − 3.67·27^-s + 1.04·29^-s − 1.49i·31^-s + 3.65i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3380$ = $2^{2} \cdot 5 \cdot 13^{2}$
Sign:	$-0.969 + 0.246i$
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:	$(2,\ 3380,\ (\ :1/2),\ -0.969 + 0.246i)$

Particular Values

$L(1)$	$\approx$	$0.3200706171$
$L(\frac12)$	$\approx$	$0.3200706171$
$L(\frac{3}{2})$		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 + 3.40T + 3T^{2} $
	7	$ 1 - 2.33iT - 7T^{2} $
	11	$ 1 + 6.16iT - 11T^{2} $
	17	$ 1 + 3.07T + 17T^{2} $
	19	$ 1 - 0.0902iT - 19T^{2} $
	23	$ 1 + 4.81T + 23T^{2} $
	29	$ 1 - 5.63T + 29T^{2} $
	31	$ 1 + 8.34iT - 31T^{2} $
	37	$ 1 - 0.794iT - 37T^{2} $
	41	$ 1 - 3.95iT - 41T^{2} $
	43	$ 1 - 8.75T + 43T^{2} $
	47	$ 1 + 3.67iT - 47T^{2} $
	53	$ 1 - 8.08T + 53T^{2} $
	59	$ 1 - 0.379iT - 59T^{2} $
	61	$ 1 + 4.96T + 61T^{2} $
	67	$ 1 + 0.376iT - 67T^{2} $
	71	$ 1 - 9.04iT - 71T^{2} $
	73	$ 1 - 5.48iT - 73T^{2} $
	79	$ 1 + 4.79T + 79T^{2} $
	83	$ 1 + 11.6iT - 83T^{2} $
	89	$ 1 + 11.7iT - 89T^{2} $
	97	$ 1 + 2.43iT - 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

−8.323036960371071825807767814645, −7.38558155973896677878480266487, −6.30736398294877889974268039408, −5.96826404838228262925941832478, −5.50765082773689627351096935664, −4.59959999839894714949926191737, −3.87416057154038200633364916422, −2.40416531439285261320137622303, −1.06555220621679893198066167340, −0.16583068435185268872144346625, 1.16108316913487203680645131877, 2.19304950867686081337091662206, 3.96083945439903249039884266989, 4.46003075208864476833407153427, 5.09775470102597593988305198461, 6.05017402739561648443626408483, 6.80162433211879204581067367921, 7.10075050892373761649413410449, 7.79409828804227680717283187744, 9.260301046624615049215412593822

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3200706171\)
\(L(\frac12)\)	\(\approx\)	\(0.3200706171\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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