LMFDB - L-function 2-4788-133.132-c1-0-66

Dirichlet series

L(s) = 1

− 4.27i·5^-s + (1.13 − 2.38i)7^-s + 2.15·11^-s − 0.274i·17^-s + 4.35i·19^-s − 8.71·23^-s − 13.2·25^-s + (−10.2 − 4.86i)35^-s − 10.8·43^-s − 10.2i·47^-s + (−4.41 − 5.43i)49^-s − 9.19i·55^-s + 15.1i·61^-s − 2.98i·73^-s + (2.44 − 5.13i)77^-s + ⋯

L(s) = 1

− 1.91i·5^-s + (0.429 − 0.902i)7^-s + 0.648·11^-s − 0.0666i·17^-s + 0.999i·19^-s − 1.81·23^-s − 2.65·25^-s + (−1.72 − 0.821i)35^-s − 1.65·43^-s − 1.49i·47^-s + (−0.630 − 0.776i)49^-s − 1.23i·55^-s + 1.94i·61^-s − 0.349i·73^-s + (0.278 − 0.585i)77^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4788$ = $2^{2} \cdot 3^{2} \cdot 7 \cdot 19$
Sign:	$-0.902 - 0.429i$
Analytic conductor:	$38.2323$
Root analytic conductor:	$6.18323$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4788} (3457, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4788,\ (\ :1/2),\ -0.902 - 0.429i)$

Particular Values

$L(1)$	$\approx$	$0.9400617574$
$L(\frac12)$	$\approx$	$0.9400617574$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (-1.13 + 2.38i)T $
	19	$ 1 - 4.35iT $
good	5	$ 1 + 4.27iT - 5T^{2} $
	11	$ 1 - 2.15T + 11T^{2} $
	13	$ 1 + 13T^{2} $
	17	$ 1 + 0.274iT - 17T^{2} $
	23	$ 1 + 8.71T + 23T^{2} $
	29	$ 1 - 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 37T^{2} $
	41	$ 1 + 41T^{2} $
	43	$ 1 + 10.8T + 43T^{2} $
	47	$ 1 + 10.2iT - 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 15.1iT - 61T^{2} $
	67	$ 1 - 67T^{2} $
	71	$ 1 - 71T^{2} $
	73	$ 1 + 2.98iT - 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 + 16iT - 83T^{2} $
	89	$ 1 + 89T^{2} $
	97	$ 1 + 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

−8.074956729001351382337881056332, −7.34154039710492284458297199739, −6.29771277070161195996649676662, −5.56845994320938185359888370407, −4.84198793863636285278349141516, −4.11869720376379473407117227168, −3.69681731720600744952169197880, −1.88940521856649130362146197349, −1.34031154422252612277520027858, −0.24376123539604450909640035407, 1.80098587634226705777557698142, 2.49839953159653315202832061915, 3.26622630577259000769840519619, 4.06119571803000183462266258181, 5.09206850597463874524555034469, 6.12876850460984176344053991510, 6.38835848113723807933510196080, 7.19070632970462574845336615407, 7.921085743830061977397748803055, 8.563225563598359289180044552178

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9400617574\)
\(L(\frac12)\)	\(\approx\)	\(0.9400617574\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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