LMFDB - L-function 2-2646-21.20-c1-0-38

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s + 2.49·5^-s + i·8^-s − 2.49i·10^-s − 0.874i·11^-s + 2.76i·13^-s + 16^-s − 2.44·17^-s − 6.86i·19^-s − 2.49·20^-s − 0.874·22^-s + 4.78i·23^-s + 1.23·25^-s + 2.76·26^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s + 1.11·5^-s + 0.353i·8^-s − 0.789i·10^-s − 0.263i·11^-s + 0.767i·13^-s + 0.250·16^-s − 0.594·17^-s − 1.57i·19^-s − 0.558·20^-s − 0.186·22^-s + 0.997i·23^-s + 0.247·25^-s + 0.542·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2646$ = $2 \cdot 3^{3} \cdot 7^{2}$
Sign:	$-0.156 + 0.987i$
Analytic conductor:	$21.1284$
Root analytic conductor:	$4.59656$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2646} (2645, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2646,\ (\ :1/2),\ -0.156 + 0.987i)$

Particular Values

$L(1)$	$\approx$	$1.941919986$
$L(\frac12)$	$\approx$	$1.941919986$
$L(\frac{3}{2})$		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 + iT $
	3	$ 1 $
	7	$ 1 $
good	5	$ 1 - 2.49T + 5T^{2} $
	11	$ 1 + 0.874iT - 11T^{2} $
	13	$ 1 - 2.76iT - 13T^{2} $
	17	$ 1 + 2.44T + 17T^{2} $
	19	$ 1 + 6.86iT - 19T^{2} $
	23	$ 1 - 4.78iT - 23T^{2} $
	29	$ 1 + 9.26iT - 29T^{2} $
	31	$ 1 + 8.64iT - 31T^{2} $
	37	$ 1 - 4.48T + 37T^{2} $
	41	$ 1 + 0.0376T + 41T^{2} $
	43	$ 1 - 5.35T + 43T^{2} $
	47	$ 1 - 4.66T + 47T^{2} $
	53	$ 1 + 10.2iT - 53T^{2} $
	59	$ 1 - 8.56T + 59T^{2} $
	61	$ 1 - 0.327iT - 61T^{2} $
	67	$ 1 - 5.72T + 67T^{2} $
	71	$ 1 + 6.54iT - 71T^{2} $
	73	$ 1 - 8.24iT - 73T^{2} $
	79	$ 1 - 16.5T + 79T^{2} $
	83	$ 1 + 11.0T + 83T^{2} $
	89	$ 1 - 8.82T + 89T^{2} $
	97	$ 1 - 8.33iT - 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

−8.952353102058469622825469552909, −8.050125686463831166331276767955, −7.06191848928346408941452186941, −6.20316429179070952926604297020, −5.55422440789417454280431869807, −4.58895114734744093066186795210, −3.83378057181361652089038987682, −2.47998309428181392967157629850, −2.10506877303444186113810491614, −0.68253008394299992891412438586, 1.21917735129411720636025523811, 2.35559761703606845631949231268, 3.47095933993778444749456414142, 4.56089121381355528778209082954, 5.40081719922791271364429731375, 5.96647254546452345248898244932, 6.69530697389747497548156746072, 7.46956015145055130187151399062, 8.369275062950268797191171622117, 8.941333541630346231725591156546

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.941919986\)
\(L(\frac12)\)	\(\approx\)	\(1.941919986\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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