LMFDB - L-function 2-2300-115.114-c2-0-40

Dirichlet series

L(s) = 1

− 1.82i·3^-s − 12.2·7^-s + 5.66·9^-s + 20.0i·11^-s + 5.97i·13^-s − 6.08·17^-s + 17.6i·19^-s + 22.4i·21^-s + (2.91 − 22.8i)23^-s − 26.7i·27^-s − 52.0·29^-s + 27.5·31^-s + 36.6·33^-s − 19.1·37^-s + 10.9·39^-s + ⋯

L(s) = 1

− 0.609i·3^-s − 1.75·7^-s + 0.629·9^-s + 1.82i·11^-s + 0.459i·13^-s − 0.358·17^-s + 0.929i·19^-s + 1.06i·21^-s + (0.126 − 0.991i)23^-s − 0.992i·27^-s − 1.79·29^-s + 0.887·31^-s + 1.11·33^-s − 0.516·37^-s + 0.280·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2300$ = $2^{2} \cdot 5^{2} \cdot 23$
Sign:	$-0.557 + 0.830i$
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:	$(2,\ 2300,\ (\ :1),\ -0.557 + 0.830i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.6094043423$
$L(\frac12)$	$\approx$	$0.6094043423$
$L(2)$		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 + (-2.91 + 22.8i)T $
good	3	$ 1 + 1.82iT - 9T^{2} $
	7	$ 1 + 12.2T + 49T^{2} $
	11	$ 1 - 20.0iT - 121T^{2} $
	13	$ 1 - 5.97iT - 169T^{2} $
	17	$ 1 + 6.08T + 289T^{2} $
	19	$ 1 - 17.6iT - 361T^{2} $
	29	$ 1 + 52.0T + 841T^{2} $
	31	$ 1 - 27.5T + 961T^{2} $
	37	$ 1 + 19.1T + 1.36e3T^{2} $
	41	$ 1 + 43.3T + 1.68e3T^{2} $
	43	$ 1 - 31.2T + 1.84e3T^{2} $
	47	$ 1 + 31.0iT - 2.20e3T^{2} $
	53	$ 1 - 42.8T + 2.80e3T^{2} $
	59	$ 1 - 99.3T + 3.48e3T^{2} $
	61	$ 1 + 91.6iT - 3.72e3T^{2} $
	67	$ 1 - 30.6T + 4.48e3T^{2} $
	71	$ 1 + 49.2T + 5.04e3T^{2} $
	73	$ 1 - 7.45iT - 5.32e3T^{2} $
	79	$ 1 - 53.0iT - 6.24e3T^{2} $
	83	$ 1 + 49.4T + 6.88e3T^{2} $
	89	$ 1 + 107. iT - 7.92e3T^{2} $
	97	$ 1 + 97.6T + 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

−8.587872490119279558704419273253, −7.52162970327768385447292168671, −6.83437949341537277189808759808, −6.63972248721794279617700855791, −5.53867579063367746189629493057, −4.35531684423697449976238393075, −3.73690362215738306834880070981, −2.45807338215960983657990728354, −1.70001958512803790013616404709, −0.17843668285081701731537717475, 0.906057760116968978684806188251, 2.68451963546874621060773269551, 3.46133875251662596446007194452, 3.95577977821612243033424099257, 5.30039820634901897777531671409, 5.89959071962399564520535991758, 6.74210969044222030983010566170, 7.41237020223054285341342277495, 8.604501519268400269021061212647, 9.167414242226621519811517876876

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.6094043423\)
\(L(\frac12)\)	\(\approx\)	\(0.6094043423\)
\(L(2)\)		not available
\(L(1)\)		not available

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