LMFDB - L-function 2-4600-5.4-c1-0-99

Dirichlet series

L(s) = 1

− 3.30i·3^-s − 2.55i·7^-s − 7.93·9^-s − 2.72·11^-s − 7.12i·13^-s + 0.924i·17^-s − 7.51·19^-s − 8.43·21^-s + i·23^-s + 16.3i·27^-s + 2.38·29^-s + 0.866·31^-s + 9.00i·33^-s + 0.352i·37^-s − 23.5·39^-s + ⋯

L(s) = 1

− 1.90i·3^-s − 0.963i·7^-s − 2.64·9^-s − 0.821·11^-s − 1.97i·13^-s + 0.224i·17^-s − 1.72·19^-s − 1.84·21^-s + 0.208i·23^-s + 3.13i·27^-s + 0.442·29^-s + 0.155·31^-s + 1.56i·33^-s + 0.0580i·37^-s − 3.77·39^-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

$L(1)$	$\approx$	$0.6450402072$
$L(\frac12)$	$\approx$	$0.6450402072$
$L(\frac{3}{2})$		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 + 3.30iT - 3T^{2} $
	7	$ 1 + 2.55iT - 7T^{2} $
	11	$ 1 + 2.72T + 11T^{2} $
	13	$ 1 + 7.12iT - 13T^{2} $
	17	$ 1 - 0.924iT - 17T^{2} $
	19	$ 1 + 7.51T + 19T^{2} $
	29	$ 1 - 2.38T + 29T^{2} $
	31	$ 1 - 0.866T + 31T^{2} $
	37	$ 1 - 0.352iT - 37T^{2} $
	41	$ 1 - 4.34T + 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 + 13.3iT - 47T^{2} $
	53	$ 1 + 3.99iT - 53T^{2} $
	59	$ 1 - 3.84T + 59T^{2} $
	61	$ 1 + 9.14T + 61T^{2} $
	67	$ 1 + 3.15iT - 67T^{2} $
	71	$ 1 + 6.07T + 71T^{2} $
	73	$ 1 - 11.3iT - 73T^{2} $
	79	$ 1 - 12.0T + 79T^{2} $
	83	$ 1 - 6.35iT - 83T^{2} $
	89	$ 1 - 9.71T + 89T^{2} $
	97	$ 1 - 8.76iT - 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

−7.65015387470986350183959337258, −7.15287660027957649908305970223, −6.37047372690210830284769644510, −5.76524503271772584468473361133, −4.98759512691188393087850484220, −3.68518160536345698928294364624, −2.78200917625476261562665684948, −2.06285935240457617939441509754, −0.892607637749322061266559706866, −0.20775168096376830664996924429, 2.17451595745106132669904686681, 2.76958646165625515992098057370, 3.84761079658672148329392944503, 4.59412653041187936159442660010, 4.86132007316205789977105347676, 6.01936428957616865021324066973, 6.31182728540208707898477703875, 7.67626493849287101686428159555, 8.617096718452230848430370268926, 9.029589993081206605779321127079

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6450402072\)
\(L(\frac12)\)	\(\approx\)	\(0.6450402072\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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