LMFDB - L-function 2-2400-40.29-c1-0-21

Dirichlet series

L(s) = 1

− 3^-s + 0.0802i·7^-s + 9^-s − 2.41i·11^-s + 5.26·13^-s + 0.255i·17^-s − 6.95i·19^-s − 0.0802i·21^-s + 1.64i·23^-s − 27^-s + 4.51i·29^-s − 8.29·31^-s + 2.41i·33^-s − 2.67·37^-s − 5.26·39^-s + ⋯

L(s) = 1

− 0.577·3^-s + 0.0303i·7^-s + 0.333·9^-s − 0.728i·11^-s + 1.46·13^-s + 0.0620i·17^-s − 1.59i·19^-s − 0.0175i·21^-s + 0.343i·23^-s − 0.192·27^-s + 0.838i·29^-s − 1.48·31^-s + 0.420i·33^-s − 0.439·37^-s − 0.843·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2400$ = $2^{5} \cdot 3 \cdot 5^{2}$
Sign:	$0.321 + 0.947i$
Analytic conductor:	$19.1640$
Root analytic conductor:	$4.37768$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2400} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2400,\ (\ :1/2),\ 0.321 + 0.947i)$

Particular Values

$L(1)$	$\approx$	$1.312012085$
$L(\frac12)$	$\approx$	$1.312012085$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 + T $
	5	$ 1 $
good	7	$ 1 - 0.0802iT - 7T^{2} $
	11	$ 1 + 2.41iT - 11T^{2} $
	13	$ 1 - 5.26T + 13T^{2} $
	17	$ 1 - 0.255iT - 17T^{2} $
	19	$ 1 + 6.95iT - 19T^{2} $
	23	$ 1 - 1.64iT - 23T^{2} $
	29	$ 1 - 4.51iT - 29T^{2} $
	31	$ 1 + 8.29T + 31T^{2} $
	37	$ 1 + 2.67T + 37T^{2} $
	41	$ 1 + 8.11T + 41T^{2} $
	43	$ 1 - 4.08T + 43T^{2} $
	47	$ 1 - 5.70iT - 47T^{2} $
	53	$ 1 - 11.5T + 53T^{2} $
	59	$ 1 + 12.6iT - 59T^{2} $
	61	$ 1 + 11.9iT - 61T^{2} $
	67	$ 1 + 7.27T + 67T^{2} $
	71	$ 1 - 11.3T + 71T^{2} $
	73	$ 1 + 12.0iT - 73T^{2} $
	79	$ 1 - 5.50T + 79T^{2} $
	83	$ 1 - 9.20T + 83T^{2} $
	89	$ 1 + 11.9T + 89T^{2} $
	97	$ 1 + 8.50iT - 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

−8.897432479844543068057651017950, −8.116729951923189825700979259501, −7.09185926627449943673027840857, −6.49637612616852531725918615631, −5.64395203386750422010877464359, −5.02553685735614893600685362638, −3.88757955843201531679843348448, −3.16719343781784131971764013680, −1.75023146926106635416250624636, −0.54972382981189535137403420212, 1.15526793554060371900493713134, 2.18240175212338566448520611142, 3.69792668127017819745938952626, 4.14022660092016714899355172096, 5.42432925960105422193825437892, 5.86332663136382769335093449670, 6.78637820736672959203771936513, 7.48758014699705561270038772865, 8.396381964479598829353337751520, 9.038853709180411902275121308037

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.312012085\)
\(L(\frac12)\)	\(\approx\)	\(1.312012085\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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