LMFDB - L-function 2-6336-33.32-c1-0-48

Dirichlet series

L(s) = 1

− 3.92i·5^-s + 2.07i·7^-s + (−3.24 − 0.665i)11^-s + 5.34i·13^-s + 0.941·17^-s − 4.59i·19^-s + 6.67i·23^-s − 10.4·25^-s + 6.61·29^-s − 7.55·31^-s + 8.17·35^-s + 3.55·37^-s + 12.0·41^-s + 1.06i·43^-s − 0.313i·47^-s + ⋯

L(s) = 1

− 1.75i·5^-s + 0.786i·7^-s + (−0.979 − 0.200i)11^-s + 1.48i·13^-s + 0.228·17^-s − 1.05i·19^-s + 1.39i·23^-s − 2.08·25^-s + 1.22·29^-s − 1.35·31^-s + 1.38·35^-s + 0.584·37^-s + 1.88·41^-s + 0.161i·43^-s − 0.0457i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6336$ = $2^{6} \cdot 3^{2} \cdot 11$
Sign:	$0.401 + 0.915i$
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:	$(2,\ 6336,\ (\ :1/2),\ 0.401 + 0.915i)$

Particular Values

$L(1)$	$\approx$	$1.592786648$
$L(\frac12)$	$\approx$	$1.592786648$
$L(\frac{3}{2})$		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 + (3.24 + 0.665i)T $
good	5	$ 1 + 3.92iT - 5T^{2} $
	7	$ 1 - 2.07iT - 7T^{2} $
	13	$ 1 - 5.34iT - 13T^{2} $
	17	$ 1 - 0.941T + 17T^{2} $
	19	$ 1 + 4.59iT - 19T^{2} $
	23	$ 1 - 6.67iT - 23T^{2} $
	29	$ 1 - 6.61T + 29T^{2} $
	31	$ 1 + 7.55T + 31T^{2} $
	37	$ 1 - 3.55T + 37T^{2} $
	41	$ 1 - 12.0T + 41T^{2} $
	43	$ 1 - 1.06iT - 43T^{2} $
	47	$ 1 + 0.313iT - 47T^{2} $
	53	$ 1 + 1.10iT - 53T^{2} $
	59	$ 1 + 12.0iT - 59T^{2} $
	61	$ 1 + 1.64iT - 61T^{2} $
	67	$ 1 + 5.88T + 67T^{2} $
	71	$ 1 + 5.34iT - 71T^{2} $
	73	$ 1 + 11.3iT - 73T^{2} $
	79	$ 1 + 10.1iT - 79T^{2} $
	83	$ 1 - 13.4T + 83T^{2} $
	89	$ 1 - 4.24iT - 89T^{2} $
	97	$ 1 - 7.67T + 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

−7.939498121782842704543248239562, −7.43332298947662193995760275724, −6.30201236688893939045225188150, −5.64780238355423022739088369239, −4.94170431935442946820585751226, −4.57586954721444586100878054524, −3.55614520374784718741859930596, −2.39752970305435452347302161993, −1.65102469789100842594069709855, −0.53126780489476157606439374559, 0.802133509981442754617012046973, 2.39235935579456141365728410345, 2.83344462001460803479159387128, 3.63749480589135947460441011605, 4.39950115723436522209377816877, 5.55650797484765563567207690057, 6.03821353848827590183104563844, 6.85636832220717662634763074318, 7.65168049855850021657717282669, 7.71120722208555371944071409279

Overview	Random
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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

\(L(1)\)	\(\approx\)	\(1.592786648\)
\(L(\frac12)\)	\(\approx\)	\(1.592786648\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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