LMFDB - L-function 2-2736-57.56-c1-0-7

Dirichlet series

L(s) = 1

− 3.42i·5^-s + 0.394·7^-s + 2.33i·11^-s + 3.36i·13^-s − 0.494i·17^-s + (0.300 + 4.34i)19^-s + 3.79i·23^-s − 6.75·25^-s + 2.14·29^-s + 7.04i·31^-s − 1.35i·35^-s + 7.30i·37^-s + 3.81·41^-s + 8.45·43^-s + 2.43i·47^-s + ⋯

L(s) = 1

− 1.53i·5^-s + 0.148·7^-s + 0.703i·11^-s + 0.933i·13^-s − 0.119i·17^-s + (0.0690 + 0.997i)19^-s + 0.790i·23^-s − 1.35·25^-s + 0.399·29^-s + 1.26i·31^-s − 0.228i·35^-s + 1.20i·37^-s + 0.595·41^-s + 1.28·43^-s + 0.355i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.774 - 0.632i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ 0.774 - 0.632i)$

Particular Values

$L(1)$	$\approx$	$1.548117145$
$L(\frac12)$	$\approx$	$1.548117145$
$L(\frac{3}{2})$		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 $
	19	$ 1 + (-0.300 - 4.34i)T $
good	5	$ 1 + 3.42iT - 5T^{2} $
	7	$ 1 - 0.394T + 7T^{2} $
	11	$ 1 - 2.33iT - 11T^{2} $
	13	$ 1 - 3.36iT - 13T^{2} $
	17	$ 1 + 0.494iT - 17T^{2} $
	23	$ 1 - 3.79iT - 23T^{2} $
	29	$ 1 - 2.14T + 29T^{2} $
	31	$ 1 - 7.04iT - 31T^{2} $
	37	$ 1 - 7.30iT - 37T^{2} $
	41	$ 1 - 3.81T + 41T^{2} $
	43	$ 1 - 8.45T + 43T^{2} $
	47	$ 1 - 2.43iT - 47T^{2} $
	53	$ 1 + 10.9T + 53T^{2} $
	59	$ 1 - 6.33T + 59T^{2} $
	61	$ 1 + 10.5T + 61T^{2} $
	67	$ 1 + 5.01iT - 67T^{2} $
	71	$ 1 - 0.759T + 71T^{2} $
	73	$ 1 - 3.06T + 73T^{2} $
	79	$ 1 - 6.40iT - 79T^{2} $
	83	$ 1 - 1.54iT - 83T^{2} $
	89	$ 1 - 13.2T + 89T^{2} $
	97	$ 1 + 1.71iT - 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

−8.978252283941157666483600688985, −8.185080802797480898078714371404, −7.57713357152765820233320487509, −6.58718373937996131531561152033, −5.72281681428625190314208986659, −4.79303425787375993177599634423, −4.47209584271183464595753415237, −3.36074612651371626150426550222, −1.88609722570786877711284702513, −1.19866591368158972790674364463, 0.54239021012702089860349712396, 2.31749561623947648162121564193, 2.94072726966695542588225919662, 3.75727936820041551364535834475, 4.82863184353351927493925882875, 5.95035812269721095932423483703, 6.33174919807929315440383564514, 7.32675213771635826924069589875, 7.78960579596512029117383956863, 8.709650990943356991366368655355

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

\(L(1)\)	\(\approx\)	\(1.548117145\)
\(L(\frac12)\)	\(\approx\)	\(1.548117145\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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