LMFDB - L-function 2-6336-33.32-c1-0-69

Dirichlet series

L(s) = 1

− 0.408i·5^-s + 1.15i·7^-s + (2.10 − 2.56i)11^-s − 1.00i·13^-s − 3.62·17^-s − 2.97i·19^-s + 4.12i·23^-s + 4.83·25^-s + 5.04·29^-s − 1.42·31^-s + 0.470·35^-s − 2.57·37^-s − 4.78·41^-s − 8.63i·43^-s + 4.65i·47^-s + ⋯

L(s) = 1

− 0.182i·5^-s + 0.434i·7^-s + (0.634 − 0.773i)11^-s − 0.278i·13^-s − 0.879·17^-s − 0.682i·19^-s + 0.859i·23^-s + 0.966·25^-s + 0.937·29^-s − 0.255·31^-s + 0.0795·35^-s − 0.423·37^-s − 0.747·41^-s − 1.31i·43^-s + 0.678i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6336$ = $2^{6} \cdot 3^{2} \cdot 11$
Sign:	$0.265 + 0.964i$
Analytic conductor:	$50.5932$
Root analytic conductor:	$7.11289$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6336} (2177, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6336,\ (\ :1/2),\ 0.265 + 0.964i)$

Particular Values

$L(1)$	$\approx$	$1.684286372$
$L(\frac12)$	$\approx$	$1.684286372$
$L(\frac{3}{2})$		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 $
	11	$ 1 + (-2.10 + 2.56i)T $
good	5	$ 1 + 0.408iT - 5T^{2} $
	7	$ 1 - 1.15iT - 7T^{2} $
	13	$ 1 + 1.00iT - 13T^{2} $
	17	$ 1 + 3.62T + 17T^{2} $
	19	$ 1 + 2.97iT - 19T^{2} $
	23	$ 1 - 4.12iT - 23T^{2} $
	29	$ 1 - 5.04T + 29T^{2} $
	31	$ 1 + 1.42T + 31T^{2} $
	37	$ 1 + 2.57T + 37T^{2} $
	41	$ 1 + 4.78T + 41T^{2} $
	43	$ 1 + 8.63iT - 43T^{2} $
	47	$ 1 - 4.65iT - 47T^{2} $
	53	$ 1 + 3.23iT - 53T^{2} $
	59	$ 1 + 3.11iT - 59T^{2} $
	61	$ 1 + 0.478iT - 61T^{2} $
	67	$ 1 - 3.25T + 67T^{2} $
	71	$ 1 - 1.00iT - 71T^{2} $
	73	$ 1 + 12.4iT - 73T^{2} $
	79	$ 1 - 11.1iT - 79T^{2} $
	83	$ 1 + 1.83T + 83T^{2} $
	89	$ 1 + 4.24iT - 89T^{2} $
	97	$ 1 - 10.6T + 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

−8.029029905074761866495554400362, −6.99162764542665613693571591287, −6.60468078322629364252468116574, −5.68569916083474689905379632135, −5.10719270112201524950100045965, −4.26388214273352947928101271632, −3.39454682609549472545293469346, −2.64521172861639067990889650945, −1.59302231456270160823573609804, −0.47585331347691975435344660789, 1.03176208697369874935708387503, 2.02232088393846415617252814350, 2.91397983294943252138661032946, 3.94509478692374355046502466371, 4.47163424618216872947335007214, 5.21025713398511547012301962174, 6.34806624867970048509972488220, 6.71396844688504107461429111439, 7.34257205093546885533354131167, 8.220573395704910190734283052788

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

\(L(1)\)	\(\approx\)	\(1.684286372\)
\(L(\frac12)\)	\(\approx\)	\(1.684286372\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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