LMFDB - L-function 2-507-13.12-c3-0-56

Dirichlet series

L(s) = 1

− 3·3^-s + 8·4^-s − 5.19i·5^-s + 10.3i·7^-s + 9·9^-s − 51.9i·11^-s − 24·12^-s + 15.5i·15^-s + 64·16^-s − 117·17^-s − 24.2i·19^-s − 41.5i·20^-s − 31.1i·21^-s + 18·23^-s + 98·25^-s + ⋯

L(s) = 1

− 0.577·3^-s + 4^-s − 0.464i·5^-s + 0.561i·7^-s + 0.333·9^-s − 1.42i·11^-s − 0.577·12^-s + 0.268i·15^-s + 16^-s − 1.66·17^-s − 0.292i·19^-s − 0.464i·20^-s − 0.323i·21^-s + 0.163·23^-s + 0.784·25^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$507$ = $3 \cdot 13^{2}$
Sign:	$-0.277 + 0.960i$
Analytic conductor:	$29.9139$
Root analytic conductor:	$5.46936$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{507} (337, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 507,\ (\ :3/2),\ -0.277 + 0.960i)$

Particular Values

$L(2)$	$\approx$	$1.450193101$
$L(\frac12)$	$\approx$	$1.450193101$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 + 3T $
	13	$ 1 $
good	2	$ 1 - 8T^{2} $
	5	$ 1 + 5.19iT - 125T^{2} $
	7	$ 1 - 10.3iT - 343T^{2} $
	11	$ 1 + 51.9iT - 1.33e3T^{2} $
	17	$ 1 + 117T + 4.91e3T^{2} $
	19	$ 1 + 24.2iT - 6.85e3T^{2} $
	23	$ 1 - 18T + 1.21e4T^{2} $
	29	$ 1 + 99T + 2.43e4T^{2} $
	31	$ 1 + 193. iT - 2.97e4T^{2} $
	37	$ 1 + 112. iT - 5.06e4T^{2} $
	41	$ 1 - 36.3iT - 6.89e4T^{2} $
	43	$ 1 + 82T + 7.95e4T^{2} $
	47	$ 1 - 72.7iT - 1.03e5T^{2} $
	53	$ 1 + 261T + 1.48e5T^{2} $
	59	$ 1 + 789. iT - 2.05e5T^{2} $
	61	$ 1 + 719T + 2.26e5T^{2} $
	67	$ 1 - 703. iT - 3.00e5T^{2} $
	71	$ 1 + 467. iT - 3.57e5T^{2} $
	73	$ 1 + 684. iT - 3.89e5T^{2} $
	79	$ 1 + 440T + 4.93e5T^{2} $
	83	$ 1 + 1.19e3iT - 5.71e5T^{2} $
	89	$ 1 + 1.51e3iT - 7.04e5T^{2} $
	97	$ 1 + 1.15e3iT - 9.12e5T^{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

−10.59435348400183086916922527739, −9.232427400492467480500414024920, −8.523804449304950013507301119182, −7.39201319710844994345600880938, −6.33221169210607497419078390249, −5.79927056157897466393676768853, −4.63445663751805936925757277499, −3.16696449929894139800470380346, −1.95192339247304154875575208710, −0.45477283372033126823421343651, 1.49165933183868599628163184016, 2.61907955212781802327804428360, 4.07515370850593451866546208597, 5.13397355008655610771556281911, 6.53130323875136454742302039664, 6.89568623477962945494789150529, 7.69352686227346422618522077820, 9.110307475348683644266951629779, 10.30434883002948537473483227863, 10.68146975157105867078297431534

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.450193101\)
\(L(\frac12)\)	\(\approx\)	\(1.450193101\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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