LMFDB - L-function 2-1148-41.40-c1-0-4

Dirichlet series

L(s) = 1

− 2.23i·3^-s + 0.468·5^-s + i·7^-s − 1.99·9^-s + 4.98i·11^-s + 3.78i·13^-s − 1.04i·15^-s + 1.22i·17^-s + 7.89i·19^-s + 2.23·21^-s − 0.249·23^-s − 4.78·25^-s − 2.24i·27^-s + 8.31i·29^-s + 3.17·31^-s + ⋯

L(s) = 1

− 1.29i·3^-s + 0.209·5^-s + 0.377i·7^-s − 0.665·9^-s + 1.50i·11^-s + 1.05i·13^-s − 0.270i·15^-s + 0.296i·17^-s + 1.81i·19^-s + 0.487·21^-s − 0.0520·23^-s − 0.956·25^-s − 0.431i·27^-s + 1.54i·29^-s + 0.569·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1148$ = $2^{2} \cdot 7 \cdot 41$
Sign:	$0.802 - 0.597i$
Analytic conductor:	$9.16682$
Root analytic conductor:	$3.02767$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1148} (1065, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1148,\ (\ :1/2),\ 0.802 - 0.597i)$

Particular Values

$L(1)$	$\approx$	$1.387092267$
$L(\frac12)$	$\approx$	$1.387092267$
$L(\frac{3}{2})$		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 $
	7	$ 1 - iT $
	41	$ 1 + (3.82 + 5.13i)T $
good	3	$ 1 + 2.23iT - 3T^{2} $
	5	$ 1 - 0.468T + 5T^{2} $
	11	$ 1 - 4.98iT - 11T^{2} $
	13	$ 1 - 3.78iT - 13T^{2} $
	17	$ 1 - 1.22iT - 17T^{2} $
	19	$ 1 - 7.89iT - 19T^{2} $
	23	$ 1 + 0.249T + 23T^{2} $
	29	$ 1 - 8.31iT - 29T^{2} $
	31	$ 1 - 3.17T + 31T^{2} $
	37	$ 1 + 7.74T + 37T^{2} $
	43	$ 1 - 4.90T + 43T^{2} $
	47	$ 1 - 1.80iT - 47T^{2} $
	53	$ 1 + 1.74iT - 53T^{2} $
	59	$ 1 - 2.25T + 59T^{2} $
	61	$ 1 + 1.09T + 61T^{2} $
	67	$ 1 + 11.8iT - 67T^{2} $
	71	$ 1 + 12.3iT - 71T^{2} $
	73	$ 1 + 2.91T + 73T^{2} $
	79	$ 1 + 3.79iT - 79T^{2} $
	83	$ 1 - 8.75T + 83T^{2} $
	89	$ 1 - 15.9iT - 89T^{2} $
	97	$ 1 - 4.29iT - 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

−9.852375228192314725886675456445, −9.007716950725950557131811853945, −8.050351609600493729040560738122, −7.34266672581008438759321480419, −6.66575817722529534342332535567, −5.90581583216583616918522670569, −4.79596108767544197891409257496, −3.65062912907856778622506033581, −1.97657702934521439670286789194, −1.71647065953927309686080366703, 0.59879344580250651441006328816, 2.72304069911264051335444357516, 3.54813652868048241823597069390, 4.49305548625053920475128626876, 5.35855303490238323324083034592, 6.11744685899681466468664960806, 7.28663234792011393347537199745, 8.331203092536848219332824775180, 8.972900306045817143123684413961, 9.892843510391728642107369018627

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.387092267\)
\(L(\frac12)\)	\(\approx\)	\(1.387092267\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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