LMFDB - L-function 2-4600-5.4-c1-0-61

Dirichlet series

L(s) = 1

− 3.07i·3^-s − 2.07i·7^-s − 6.48·9^-s + 5.07·11^-s + 3.48i·13^-s + 6.48i·17^-s + 7.48·19^-s − 6.40·21^-s − i·23^-s + 10.7i·27^-s + 1.56·29^-s + 0.0791·31^-s − 15.6i·33^-s − 9.71i·37^-s + 10.7·39^-s + ⋯

L(s) = 1

− 1.77i·3^-s − 0.785i·7^-s − 2.16·9^-s + 1.53·11^-s + 0.965i·13^-s + 1.57i·17^-s + 1.71·19^-s − 1.39·21^-s − 0.208i·23^-s + 2.06i·27^-s + 0.289·29^-s + 0.0142·31^-s − 2.72i·33^-s − 1.59i·37^-s + 1.71·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4600$ = $2^{3} \cdot 5^{2} \cdot 23$
Sign:	$-0.447 + 0.894i$
Analytic conductor:	$36.7311$
Root analytic conductor:	$6.06062$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4600} (4049, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4600,\ (\ :1/2),\ -0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$2.176958687$
$L(\frac12)$	$\approx$	$2.176958687$
$L(\frac{3}{2})$		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 + 3.07iT - 3T^{2} $
	7	$ 1 + 2.07iT - 7T^{2} $
	11	$ 1 - 5.07T + 11T^{2} $
	13	$ 1 - 3.48iT - 13T^{2} $
	17	$ 1 - 6.48iT - 17T^{2} $
	19	$ 1 - 7.48T + 19T^{2} $
	29	$ 1 - 1.56T + 29T^{2} $
	31	$ 1 - 0.0791T + 31T^{2} $
	37	$ 1 + 9.71iT - 37T^{2} $
	41	$ 1 + 0.480T + 41T^{2} $
	43	$ 1 + 8iT - 43T^{2} $
	47	$ 1 - 6.96iT - 47T^{2} $
	53	$ 1 + 11.7iT - 53T^{2} $
	59	$ 1 - 11.5T + 59T^{2} $
	61	$ 1 - 7.88T + 61T^{2} $
	67	$ 1 + 9.71iT - 67T^{2} $
	71	$ 1 + 9.67T + 71T^{2} $
	73	$ 1 - 13.2iT - 73T^{2} $
	79	$ 1 - 12.3T + 79T^{2} $
	83	$ 1 - 4.59iT - 83T^{2} $
	89	$ 1 + 8.31T + 89T^{2} $
	97	$ 1 - 7.23iT - 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

−7.953417865591742137987763846865, −7.07562599537961612120530400612, −6.91630488626717906698764013698, −6.15913775270643562278263504133, −5.43841758223119210162516531326, −4.07876165590233436536087369665, −3.56210284994336987360116243772, −2.21204871927215847940364126987, −1.47575122214460229612176742393, −0.78827300348189271728511041563, 0.996754080179682520150237908565, 2.76742338734489608704131622068, 3.20765812373988208530851551231, 4.05187724026678254465706697962, 4.95037052750560516897094422676, 5.34681743300551116413436745076, 6.10917256979237677979547879655, 7.09257529941082758849995709858, 8.047941271962651953959156250626, 8.935996983105362456161937020306

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.176958687\)
\(L(\frac12)\)	\(\approx\)	\(2.176958687\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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