LMFDB - L-function 2-525-5.4-c5-0-15

Dirichlet series

L(s) = 1

− 7.14i·2^-s − 9i·3^-s − 19.0·4^-s − 64.3·6^-s − 49i·7^-s − 92.5i·8^-s − 81·9^-s − 529.·11^-s + 171. i·12^-s + 1.14e3i·13^-s − 350.·14^-s − 1.27e3·16^-s − 1.11e3i·17^-s + 578. i·18^-s − 169.·19^-s + ⋯

L(s) = 1

− 1.26i·2^-s − 0.577i·3^-s − 0.595·4^-s − 0.729·6^-s − 0.377i·7^-s − 0.511i·8^-s − 0.333·9^-s − 1.32·11^-s + 0.343i·12^-s + 1.88i·13^-s − 0.477·14^-s − 1.24·16^-s − 0.939i·17^-s + 0.420i·18^-s − 0.107·19^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$525$ = $3 \cdot 5^{2} \cdot 7$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$84.2015$
Root analytic conductor:	$9.17613$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{525} (274, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 525,\ (\ :5/2),\ 0.894 + 0.447i)$

Particular Values

$L(3)$	$\approx$	$0.9142386437$
$L(\frac12)$	$\approx$	$0.9142386437$
$L(\frac{7}{2})$		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 + 9iT $
	5	$ 1 $
	7	$ 1 + 49iT $
good	2	$ 1 + 7.14iT - 32T^{2} $
	11	$ 1 + 529.T + 1.61e5T^{2} $
	13	$ 1 - 1.14e3iT - 3.71e5T^{2} $
	17	$ 1 + 1.11e3iT - 1.41e6T^{2} $
	19	$ 1 + 169.T + 2.47e6T^{2} $
	23	$ 1 + 948. iT - 6.43e6T^{2} $
	29	$ 1 - 3.26e3T + 2.05e7T^{2} $
	31	$ 1 + 2.20e3T + 2.86e7T^{2} $
	37	$ 1 - 66.9iT - 6.93e7T^{2} $
	41	$ 1 + 9.01e3T + 1.15e8T^{2} $
	43	$ 1 - 4.43e3iT - 1.47e8T^{2} $
	47	$ 1 - 2.61e4iT - 2.29e8T^{2} $
	53	$ 1 + 2.45e4iT - 4.18e8T^{2} $
	59	$ 1 - 2.90e4T + 7.14e8T^{2} $
	61	$ 1 - 3.72e4T + 8.44e8T^{2} $
	67	$ 1 - 4.65e4iT - 1.35e9T^{2} $
	71	$ 1 + 2.95e4T + 1.80e9T^{2} $
	73	$ 1 - 7.16e4iT - 2.07e9T^{2} $
	79	$ 1 - 8.01e4T + 3.07e9T^{2} $
	83	$ 1 - 5.08e4iT - 3.93e9T^{2} $
	89	$ 1 - 1.20e5T + 5.58e9T^{2} $
	97	$ 1 - 1.25e5iT - 8.58e9T^{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

−10.14087515047214720414140439701, −9.404114757322246982550954288122, −8.358515646971989228429799024139, −7.18499081323116890955256666035, −6.58166926622583567486932086472, −5.04803107720463979006981886871, −4.05377885045608490278586170244, −2.79086312439363875614255000196, −2.05607382475955297268413901274, −0.924973530722190663357820411004, 0.23423928429574645888724322563, 2.33148844083656623959586195387, 3.44661155058864436051179990744, 5.04458557610479813960066878500, 5.44974241085706121292797203376, 6.33184295245406720373876216198, 7.62293134396077913006331787896, 8.141441803412383990846163131548, 8.894517562014545752712408438632, 10.31770004360358165676120661574

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(0.9142386437\)
\(L(\frac12)\)	\(\approx\)	\(0.9142386437\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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