LMFDB - L-function 2-2300-115.114-c2-0-51

Dirichlet series

L(s) = 1

− 1.54i·3^-s + 7.17·7^-s + 6.62·9^-s + 14.0i·11^-s − 20.3i·13^-s + 20.3·17^-s + 0.231i·19^-s − 11.0i·21^-s + (−22.0 − 6.58i)23^-s − 24.0i·27^-s + 40.6·29^-s − 0.922·31^-s + 21.6·33^-s + 7.73·37^-s − 31.4·39^-s + ⋯

L(s) = 1

− 0.514i·3^-s + 1.02·7^-s + 0.735·9^-s + 1.27i·11^-s − 1.56i·13^-s + 1.19·17^-s + 0.0122i·19^-s − 0.526i·21^-s + (−0.958 − 0.286i)23^-s − 0.892i·27^-s + 1.40·29^-s − 0.0297·31^-s + 0.656·33^-s + 0.209·37^-s − 0.806·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2300$ = $2^{2} \cdot 5^{2} \cdot 23$
Sign:	$0.728 + 0.684i$
Analytic conductor:	$62.6704$
Root analytic conductor:	$7.91646$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2300} (1149, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2300,\ (\ :1),\ 0.728 + 0.684i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.772480940$
$L(\frac12)$	$\approx$	$2.772480940$
$L(2)$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	5	$ 1 $
	23	$ 1 + (22.0 + 6.58i)T $
good	3	$ 1 + 1.54iT - 9T^{2} $
	7	$ 1 - 7.17T + 49T^{2} $
	11	$ 1 - 14.0iT - 121T^{2} $
	13	$ 1 + 20.3iT - 169T^{2} $
	17	$ 1 - 20.3T + 289T^{2} $
	19	$ 1 - 0.231iT - 361T^{2} $
	29	$ 1 - 40.6T + 841T^{2} $
	31	$ 1 + 0.922T + 961T^{2} $
	37	$ 1 - 7.73T + 1.36e3T^{2} $
	41	$ 1 + 1.20T + 1.68e3T^{2} $
	43	$ 1 + 17.6T + 1.84e3T^{2} $
	47	$ 1 - 14.9iT - 2.20e3T^{2} $
	53	$ 1 - 58.6T + 2.80e3T^{2} $
	59	$ 1 + 16.7T + 3.48e3T^{2} $
	61	$ 1 + 39.5iT - 3.72e3T^{2} $
	67	$ 1 - 45.2T + 4.48e3T^{2} $
	71	$ 1 - 43.5T + 5.04e3T^{2} $
	73	$ 1 - 60.0iT - 5.32e3T^{2} $
	79	$ 1 - 49.3iT - 6.24e3T^{2} $
	83	$ 1 + 56.4T + 6.88e3T^{2} $
	89	$ 1 - 65.3iT - 7.92e3T^{2} $
	97	$ 1 - 128.T + 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

−8.353094759557675811370392901341, −7.946381588810177065067740952888, −7.37032581398724035221521547546, −6.51936754301590133464232823596, −5.48470065329273473653013093326, −4.81299379689780770725729455459, −3.95568763628465634951134455938, −2.69924760371940111815250664175, −1.71216925638258249384914277880, −0.842085431093355362889068479898, 1.03195638189393860983228717521, 1.95234849866871955068183013717, 3.32210442519056194926916314601, 4.14628038128036105640558141146, 4.81972574586219083196469778990, 5.70022554697702416566913694884, 6.56943495148833088053035231377, 7.47017132504788509210725073580, 8.250856415502272320306394353015, 8.853855733523981321540835591084

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.772480940\)
\(L(\frac12)\)	\(\approx\)	\(2.772480940\)
\(L(2)\)		not available
\(L(1)\)		not available

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