LMFDB - L-function 2-624-13.12-c3-0-21

Dirichlet series

L(s) = 1

− 3·3^-s + 9.65i·5^-s + 22.3i·7^-s + 9·9^-s − 50.3i·11^-s + (−39.7 − 24.8i)13^-s − 28.9i·15^-s − 86.1·17^-s − 116. i·19^-s − 67.0i·21^-s + 72·23^-s + 31.7·25^-s − 27·27^-s + 14.1·29^-s + 196. i·31^-s + ⋯

L(s) = 1

− 0.577·3^-s + 0.863i·5^-s + 1.20i·7^-s + 0.333·9^-s − 1.37i·11^-s + (−0.847 − 0.531i)13^-s − 0.498i·15^-s − 1.22·17^-s − 1.41i·19^-s − 0.697i·21^-s + 0.652·23^-s + 0.253·25^-s − 0.192·27^-s + 0.0905·29^-s + 1.13i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.847 + 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$624$ = $2^{4} \cdot 3 \cdot 13$
Sign:	$0.847 + 0.531i$
Analytic conductor:	$36.8171$
Root analytic conductor:	$6.06771$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{624} (337, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 624,\ (\ :3/2),\ 0.847 + 0.531i)$

Particular Values

$L(2)$	$\approx$	$1.134570694$
$L(\frac12)$	$\approx$	$1.134570694$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 + 3T $
	13	$ 1 + (39.7 + 24.8i)T $
good	5	$ 1 - 9.65iT - 125T^{2} $
	7	$ 1 - 22.3iT - 343T^{2} $
	11	$ 1 + 50.3iT - 1.33e3T^{2} $
	17	$ 1 + 86.1T + 4.91e3T^{2} $
	19	$ 1 + 116. iT - 6.85e3T^{2} $
	23	$ 1 - 72T + 1.21e4T^{2} $
	29	$ 1 - 14.1T + 2.43e4T^{2} $
	31	$ 1 - 196. iT - 2.97e4T^{2} $
	37	$ 1 - 154. iT - 5.06e4T^{2} $
	41	$ 1 + 265. iT - 6.89e4T^{2} $
	43	$ 1 - 211.T + 7.95e4T^{2} $
	47	$ 1 - 67.5iT - 1.03e5T^{2} $
	53	$ 1 - 686.T + 1.48e5T^{2} $
	59	$ 1 + 91.9iT - 2.05e5T^{2} $
	61	$ 1 - 329.T + 2.26e5T^{2} $
	67	$ 1 + 768. iT - 3.00e5T^{2} $
	71	$ 1 + 264. iT - 3.57e5T^{2} $
	73	$ 1 + 771. iT - 3.89e5T^{2} $
	79	$ 1 + 1.22e3T + 4.93e5T^{2} $
	83	$ 1 - 514. iT - 5.71e5T^{2} $
	89	$ 1 + 527. iT - 7.04e5T^{2} $
	97	$ 1 + 74.2iT - 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

−10.40822178625826059132325711215, −9.090136808274251818012909598574, −8.618075107916460395022387761272, −7.21605549954595346516232100556, −6.52458451392073934965329324215, −5.60326446300382633811894292562, −4.79418730344071520372774672993, −3.13900691953855585867294867529, −2.42746807941748610970659952748, −0.45458263623674012964068728426, 0.908052569580077579108640718754, 2.12142288519310353278334880561, 4.22556313521434607967346895397, 4.45378985298353026462057350218, 5.62629711219449664054645075919, 6.96081526191463969324178481706, 7.34646249675168893059896226387, 8.568493073703884740555135784023, 9.656447093637378616413796378706, 10.15504915044611124620382389328

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.134570694\)
\(L(\frac12)\)	\(\approx\)	\(1.134570694\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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