LMFDB - L-function 2-45e2-5.4-c1-0-65

Dirichlet series

L(s) = 1

− 1.47i·2^-s − 0.170·4^-s − 3.86i·7^-s − 2.69i·8^-s + 0.260·11^-s − 4.07i·13^-s − 5.69·14^-s − 4.31·16^-s − 3.26i·17^-s − 4.24·19^-s − 0.383i·22^-s + 8.69i·23^-s − 6.00·26^-s + 0.659i·28^-s − 4.22·29^-s + ⋯

L(s) = 1

− 1.04i·2^-s − 0.0852·4^-s − 1.46i·7^-s − 0.952i·8^-s + 0.0784·11^-s − 1.13i·13^-s − 1.52·14^-s − 1.07·16^-s − 0.790i·17^-s − 0.974·19^-s − 0.0817i·22^-s + 1.81i·23^-s − 1.17·26^-s + 0.124i·28^-s − 0.784·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2025$ = $3^{4} \cdot 5^{2}$
Sign:	$-0.894 - 0.447i$
Analytic conductor:	$16.1697$
Root analytic conductor:	$4.02115$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2025} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2025,\ (\ :1/2),\ -0.894 - 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.466436788$
$L(\frac12)$	$\approx$	$1.466436788$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	2	$ 1 + 1.47iT - 2T^{2} $
	7	$ 1 + 3.86iT - 7T^{2} $
	11	$ 1 - 0.260T + 11T^{2} $
	13	$ 1 + 4.07iT - 13T^{2} $
	17	$ 1 + 3.26iT - 17T^{2} $
	19	$ 1 + 4.24T + 19T^{2} $
	23	$ 1 - 8.69iT - 23T^{2} $
	29	$ 1 + 4.22T + 29T^{2} $
	31	$ 1 - 2.65T + 31T^{2} $
	37	$ 1 - 2.27iT - 37T^{2} $
	41	$ 1 - 5.64T + 41T^{2} $
	43	$ 1 + 9.07iT - 43T^{2} $
	47	$ 1 - 1.42iT - 47T^{2} $
	53	$ 1 - 11.3iT - 53T^{2} $
	59	$ 1 - 7.12T + 59T^{2} $
	61	$ 1 - 2.52T + 61T^{2} $
	67	$ 1 + 11.2iT - 67T^{2} $
	71	$ 1 + 8.38T + 71T^{2} $
	73	$ 1 - 0.403iT - 73T^{2} $
	79	$ 1 - 3.04T + 79T^{2} $
	83	$ 1 - 4.58iT - 83T^{2} $
	89	$ 1 + 7.17T + 89T^{2} $
	97	$ 1 + 3.11iT - 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

−8.953694826563806891011165942264, −7.66668583036315218874061079988, −7.36396401291242652009700492358, −6.41061732402380268302528231975, −5.36135708978285392941889694962, −4.21959975847984711141685893203, −3.61077384732845770049674716242, −2.73110321660343213869172632412, −1.49652722679454975153939088220, −0.50019726831901561547232720397, 1.98300215446278157705738322319, 2.57528527077166780660866141920, 4.13557944412997154348215790111, 4.96842753545604949827840505196, 5.96199641891190165829089668757, 6.32675186857784347398894114534, 7.06346483874056700139390371956, 8.190198717666176789043265374065, 8.602127775648270341337726296319, 9.189160259021475472297795832871

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.466436788\)
\(L(\frac12)\)	\(\approx\)	\(1.466436788\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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