LMFDB - L-function 2-525-7.6-c2-0-20

Dirichlet series

L(s) = 1

− 2.91·2^-s − 1.73i·3^-s + 4.51·4^-s + 5.05i·6^-s + (−6.13 + 3.36i)7^-s − 1.49·8^-s − 2.99·9^-s − 2.58·11^-s − 7.81i·12^-s − 0.0498i·13^-s + (17.9 − 9.80i)14^-s − 13.6·16^-s + 14.2i·17^-s + 8.75·18^-s − 14.9i·19^-s + ⋯

L(s) = 1

− 1.45·2^-s − 0.577i·3^-s + 1.12·4^-s + 0.842i·6^-s + (−0.877 + 0.480i)7^-s − 0.186·8^-s − 0.333·9^-s − 0.235·11^-s − 0.651i·12^-s − 0.00383i·13^-s + (1.27 − 0.700i)14^-s − 0.855·16^-s + 0.835i·17^-s + 0.486·18^-s − 0.785i·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$525$ = $3 \cdot 5^{2} \cdot 7$
Sign:	$0.480 + 0.877i$
Analytic conductor:	$14.3052$
Root analytic conductor:	$3.78222$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{525} (76, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 525,\ (\ :1),\ 0.480 + 0.877i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.5641759662$
$L(\frac12)$	$\approx$	$0.5641759662$
$L(2)$		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 + 1.73iT $
	5	$ 1 $
	7	$ 1 + (6.13 - 3.36i)T $
good	2	$ 1 + 2.91T + 4T^{2} $
	11	$ 1 + 2.58T + 121T^{2} $
	13	$ 1 + 0.0498iT - 169T^{2} $
	17	$ 1 - 14.2iT - 289T^{2} $
	19	$ 1 + 14.9iT - 361T^{2} $
	23	$ 1 - 22.2T + 529T^{2} $
	29	$ 1 - 17.4T + 841T^{2} $
	31	$ 1 - 6.36iT - 961T^{2} $
	37	$ 1 - 7.14T + 1.36e3T^{2} $
	41	$ 1 - 74.5iT - 1.68e3T^{2} $
	43	$ 1 + 79.2T + 1.84e3T^{2} $
	47	$ 1 + 81.3iT - 2.20e3T^{2} $
	53	$ 1 - 67.1T + 2.80e3T^{2} $
	59	$ 1 - 4.33iT - 3.48e3T^{2} $
	61	$ 1 + 109. iT - 3.72e3T^{2} $
	67	$ 1 - 49.1T + 4.48e3T^{2} $
	71	$ 1 - 97.3T + 5.04e3T^{2} $
	73	$ 1 + 116. iT - 5.32e3T^{2} $
	79	$ 1 - 98.8T + 6.24e3T^{2} $
	83	$ 1 + 112. iT - 6.88e3T^{2} $
	89	$ 1 + 77.7iT - 7.92e3T^{2} $
	97	$ 1 - 109. iT - 9.40e3T^{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

−10.27409247604114433572644794553, −9.523352450212197216635260631839, −8.681013349543826991695419966224, −8.085362288238331165842345271820, −6.93646020321522404766186331281, −6.42135363320386757810538496162, −5.00560522517619022579059211361, −3.18878679345852237071192177269, −1.96691416391757706518426729135, −0.52236974152041496245496836047, 0.840529763803890110172943041868, 2.62996992383073959962253468278, 3.92231606271689113774808079636, 5.23149845966648981888884514230, 6.59731859731104125063473104519, 7.36130261871169232398238463866, 8.351775317042265901812708020329, 9.198071629659464385600246540679, 9.852215834732342371467606698564, 10.45476539085622565616569435079

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.5641759662\)
\(L(\frac12)\)	\(\approx\)	\(0.5641759662\)
\(L(2)\)		not available
\(L(1)\)		not available

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