LMFDB - L-function 2-2300-5.4-c1-0-17

Dirichlet series

L(s) = 1

− 2.50i·3^-s + 0.797i·7^-s − 3.28·9^-s + 4.21·11^-s + 1.48i·13^-s + 3.01i·17^-s + 2.21·19^-s + 1.99·21^-s − i·23^-s + 0.708i·27^-s + 2.57·29^-s + 8.78·31^-s − 10.5i·33^-s + 9.57i·37^-s + 3.72·39^-s + ⋯

L(s) = 1

− 1.44i·3^-s + 0.301i·7^-s − 1.09·9^-s + 1.27·11^-s + 0.411i·13^-s + 0.730i·17^-s + 0.508·19^-s + 0.436·21^-s − 0.208i·23^-s + 0.136i·27^-s + 0.477·29^-s + 1.57·31^-s − 1.83i·33^-s + 1.57i·37^-s + 0.595·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2300$ = $2^{2} \cdot 5^{2} \cdot 23$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$18.3655$
Root analytic conductor:	$4.28550$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2300} (1749, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2300,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$2.002808536$
$L(\frac12)$	$\approx$	$2.002808536$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	23	$ 1 + iT $
good	3	$ 1 + 2.50iT - 3T^{2} $
	7	$ 1 - 0.797iT - 7T^{2} $
	11	$ 1 - 4.21T + 11T^{2} $
	13	$ 1 - 1.48iT - 13T^{2} $
	17	$ 1 - 3.01iT - 17T^{2} $
	19	$ 1 - 2.21T + 19T^{2} $
	29	$ 1 - 2.57T + 29T^{2} $
	31	$ 1 - 8.78T + 31T^{2} $
	37	$ 1 - 9.57iT - 37T^{2} $
	41	$ 1 + 1.26T + 41T^{2} $
	43	$ 1 - 2.21iT - 43T^{2} $
	47	$ 1 + 2.86iT - 47T^{2} $
	53	$ 1 + 5.41iT - 53T^{2} $
	59	$ 1 - 1.21T + 59T^{2} $
	61	$ 1 - 2.58T + 61T^{2} $
	67	$ 1 + 8.56iT - 67T^{2} $
	71	$ 1 - 6.73T + 71T^{2} $
	73	$ 1 + 3.02iT - 73T^{2} $
	79	$ 1 + 0.189T + 79T^{2} $
	83	$ 1 - 7.22iT - 83T^{2} $
	89	$ 1 + 6.38T + 89T^{2} $
	97	$ 1 + 13.3iT - 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

−8.576563867092597646549002287757, −8.168894873507163194142119993018, −7.20371063602127539245582240494, −6.47004311479420072823630919678, −6.21340627873157650265194428278, −4.96432620359943642256492610975, −3.93087934010871847735702167261, −2.80251862532841403142290329846, −1.76392476073156748658557601635, −0.972767393573252336403373627329, 0.976781384595847800173710803267, 2.66765525970923060094549396694, 3.64448590660681611224104010985, 4.23330772646800621090564389980, 5.01893502412742225747021414503, 5.83828466531889919561046982320, 6.79870513799262529613221037460, 7.62785821055731766580786641494, 8.705860609714280295033896217982, 9.219324518973537464071530988878

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.002808536\)
\(L(\frac12)\)	\(\approx\)	\(2.002808536\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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