LMFDB - L-function 2-2450-5.4-c1-0-0

Dirichlet series

L(s) = 1

− i·2^-s + 3i·3^-s − 4^-s + 3·6^-s + i·8^-s − 6·9^-s − 2·11^-s − 3i·12^-s + 16^-s + 4i·17^-s + 6i·18^-s + 6·19^-s + 2i·22^-s + 3i·23^-s − 3·24^-s + ⋯

L(s) = 1

− 0.707i·2^-s + 1.73i·3^-s − 0.5·4^-s + 1.22·6^-s + 0.353i·8^-s − 2·9^-s − 0.603·11^-s − 0.866i·12^-s + 0.250·16^-s + 0.970i·17^-s + 1.41i·18^-s + 1.37·19^-s + 0.426i·22^-s + 0.625i·23^-s − 0.612·24^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2450$ = $2 \cdot 5^{2} \cdot 7^{2}$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$19.5633$
Root analytic conductor:	$4.42304$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2450} (99, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2450,\ (\ :1/2),\ -0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$0.3368947870$
$L(\frac12)$	$\approx$	$0.3368947870$
$L(\frac{3}{2})$		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 + iT $
	5	$ 1 $
	7	$ 1 $
good	3	$ 1 - 3iT - 3T^{2} $
	11	$ 1 + 2T + 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 4iT - 17T^{2} $
	19	$ 1 - 6T + 19T^{2} $
	23	$ 1 - 3iT - 23T^{2} $
	29	$ 1 + 9T + 29T^{2} $
	31	$ 1 + 4T + 31T^{2} $
	37	$ 1 - 4iT - 37T^{2} $
	41	$ 1 + 7T + 41T^{2} $
	43	$ 1 + 5iT - 43T^{2} $
	47	$ 1 + 8iT - 47T^{2} $
	53	$ 1 + 2iT - 53T^{2} $
	59	$ 1 + 10T + 59T^{2} $
	61	$ 1 - T + 61T^{2} $
	67	$ 1 - 9iT - 67T^{2} $
	71	$ 1 - 2T + 71T^{2} $
	73	$ 1 + 4iT - 73T^{2} $
	79	$ 1 + 10T + 79T^{2} $
	83	$ 1 + 7iT - 83T^{2} $
	89	$ 1 + T + 89T^{2} $
	97	$ 1 + 14iT - 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

−9.551376475644920949317094753199, −8.935990719661259047961201329530, −8.182304880898049368889721690946, −7.23388523538313548028243484496, −5.69002718408736039276116007517, −5.37529989141844816863279257437, −4.47090707626934530569205143504, −3.58903939073236119838101542511, −3.18762517810985021672298877012, −1.81999077786162835530591016042, 0.11442797233236335246687668959, 1.30922243282634210104017822535, 2.45905658221863249662093823224, 3.40295091403241655453408914145, 4.90013013610484023793098619359, 5.61132289907456208918930119980, 6.29640746464056527390283525444, 7.24750453353937110079891465061, 7.47065033358485359870642212617, 8.136525531681965881330905095430

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3368947870\)
\(L(\frac12)\)	\(\approx\)	\(0.3368947870\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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