LMFDB - L-function 2-1275-85.84-c1-0-49

Dirichlet series

L(s) = 1

− 2.67i·2^-s − 3^-s − 5.14·4^-s + 2.67i·6^-s + 3.95·7^-s + 8.39i·8^-s + 9^-s − 2.28i·11^-s + 5.14·12^-s − 1.90i·13^-s − 10.5i·14^-s + 12.1·16^-s + (−2.31 + 3.40i)17^-s − 2.67i·18^-s − 6.58·19^-s + ⋯

L(s) = 1

− 1.88i·2^-s − 0.577·3^-s − 2.57·4^-s + 1.09i·6^-s + 1.49·7^-s + 2.96i·8^-s + 0.333·9^-s − 0.689i·11^-s + 1.48·12^-s − 0.527i·13^-s − 2.82i·14^-s + 3.03·16^-s + (−0.562 + 0.826i)17^-s − 0.629i·18^-s − 1.51·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1275$ = $3 \cdot 5^{2} \cdot 17$
Sign:	$0.133 - 0.991i$
Analytic conductor:	$10.1809$
Root analytic conductor:	$3.19075$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1275} (424, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1275,\ (\ :1/2),\ 0.133 - 0.991i)$

Particular Values

$L(1)$	$\approx$	$0.4683222236$
$L(\frac12)$	$\approx$	$0.4683222236$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 + T $
	5	$ 1 $
	17	$ 1 + (2.31 - 3.40i)T $
good	2	$ 1 + 2.67iT - 2T^{2} $
	7	$ 1 - 3.95T + 7T^{2} $
	11	$ 1 + 2.28iT - 11T^{2} $
	13	$ 1 + 1.90iT - 13T^{2} $
	19	$ 1 + 6.58T + 19T^{2} $
	23	$ 1 + 7.11T + 23T^{2} $
	29	$ 1 + 2.23iT - 29T^{2} $
	31	$ 1 + 9.45iT - 31T^{2} $
	37	$ 1 + 6.57T + 37T^{2} $
	41	$ 1 - 1.52iT - 41T^{2} $
	43	$ 1 + 4.26iT - 43T^{2} $
	47	$ 1 - 3.61iT - 47T^{2} $
	53	$ 1 - 1.48iT - 53T^{2} $
	59	$ 1 + 10.7T + 59T^{2} $
	61	$ 1 + 15.0iT - 61T^{2} $
	67	$ 1 - 4.91iT - 67T^{2} $
	71	$ 1 - 9.15iT - 71T^{2} $
	73	$ 1 - 7.94T + 73T^{2} $
	79	$ 1 + 3.92iT - 79T^{2} $
	83	$ 1 - 3.85iT - 83T^{2} $
	89	$ 1 - 4.71T + 89T^{2} $
	97	$ 1 + 7.70T + 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

−9.308315481455694046764697799627, −8.243921573256570063496851118784, −8.084586203463641118387980766842, −6.19475045820600026177680256555, −5.32825470273483015450305217220, −4.36488385972982692674683305885, −3.87323194306702683297771751800, −2.33156504911571404396102036005, −1.63126822085772833700185833170, −0.21512678254384284887278644242, 1.75786402135501101210817231407, 4.13212963378144808924825769043, 4.71155515140613551495444243984, 5.28046553356398521325424420684, 6.33108897614731870431311652647, 6.95708568469706470117698405735, 7.68742928552838957021151212193, 8.484121505082716792325928579351, 9.035395344559115611194275221172, 10.13140350559918712187170101495

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

\(L(1)\)	\(\approx\)	\(0.4683222236\)
\(L(\frac12)\)	\(\approx\)	\(0.4683222236\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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