LMFDB - L-function 2-2808-13.12-c1-0-55

Dirichlet series

L(s) = 1

− 3.85i·5^-s − 2.37i·7^-s + 0.733i·11^-s + (−2.87 + 2.17i)13^-s − 4.14·17^-s − 1.41i·19^-s − 5.23·23^-s − 9.83·25^-s + 1.20·29^-s + 5.01i·31^-s − 9.16·35^-s + 10.8i·37^-s − 4.46i·41^-s + 0.930·43^-s − 6.57i·47^-s + ⋯

L(s) = 1

− 1.72i·5^-s − 0.899i·7^-s + 0.221i·11^-s + (−0.797 + 0.602i)13^-s − 1.00·17^-s − 0.323i·19^-s − 1.09·23^-s − 1.96·25^-s + 0.224·29^-s + 0.900i·31^-s − 1.54·35^-s + 1.77i·37^-s − 0.696i·41^-s + 0.141·43^-s − 0.959i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.602 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2808$ = $2^{3} \cdot 3^{3} \cdot 13$
Sign:	$-0.602 - 0.797i$
Analytic conductor:	$22.4219$
Root analytic conductor:	$4.73518$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2808} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2808,\ (\ :1/2),\ -0.602 - 0.797i)$

Particular Values

$L(1)$	$\approx$	$0.2468383182$
$L(\frac12)$	$\approx$	$0.2468383182$
$L(\frac{3}{2})$		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 $
	13	$ 1 + (2.87 - 2.17i)T $
good	5	$ 1 + 3.85iT - 5T^{2} $
	7	$ 1 + 2.37iT - 7T^{2} $
	11	$ 1 - 0.733iT - 11T^{2} $
	17	$ 1 + 4.14T + 17T^{2} $
	19	$ 1 + 1.41iT - 19T^{2} $
	23	$ 1 + 5.23T + 23T^{2} $
	29	$ 1 - 1.20T + 29T^{2} $
	31	$ 1 - 5.01iT - 31T^{2} $
	37	$ 1 - 10.8iT - 37T^{2} $
	41	$ 1 + 4.46iT - 41T^{2} $
	43	$ 1 - 0.930T + 43T^{2} $
	47	$ 1 + 6.57iT - 47T^{2} $
	53	$ 1 - 4.81T + 53T^{2} $
	59	$ 1 + 0.634iT - 59T^{2} $
	61	$ 1 + 7.96T + 61T^{2} $
	67	$ 1 + 5.70iT - 67T^{2} $
	71	$ 1 - 7.09iT - 71T^{2} $
	73	$ 1 - 7.60iT - 73T^{2} $
	79	$ 1 + 13.5T + 79T^{2} $
	83	$ 1 + 14.4iT - 83T^{2} $
	89	$ 1 - 4.41iT - 89T^{2} $
	97	$ 1 + 4.09iT - 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

−8.573720047906466440565745216518, −7.55268061812986618645705538976, −6.91693266344505233652248992798, −5.94508504905625753535883335468, −4.81129361399146413734188035039, −4.62024700235110221330739723969, −3.74274301119714469474339236657, −2.22677554122599879025906094638, −1.26006934137301020118809823624, −0.07750785450032449568575404426, 2.18160540469291970290273075025, 2.60525703275603645938592149964, 3.54901754696182350000497363093, 4.52441597882435092343064956439, 5.89210178221485237842826520855, 6.00735529881049972672617179915, 7.10706949846454643531208301910, 7.61843840524685960110914018916, 8.466365234116549868062158622993, 9.406246507980587948523004577697

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.2468383182\)
\(L(\frac12)\)	\(\approx\)	\(0.2468383182\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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