LMFDB - L-function 2-30e2-5.4-c3-0-13

Dirichlet series

L(s) = 1

− 2i·7^-s − 30·11^-s − 4i·13^-s + 90i·17^-s + 28·19^-s − 120i·23^-s + 210·29^-s − 4·31^-s − 200i·37^-s − 240·41^-s − 136i·43^-s − 120i·47^-s + 339·49^-s + 30i·53^-s − 450·59^-s + ⋯

L(s) = 1

− 0.107i·7^-s − 0.822·11^-s − 0.0853i·13^-s + 1.28i·17^-s + 0.338·19^-s − 1.08i·23^-s + 1.34·29^-s − 0.0231·31^-s − 0.888i·37^-s − 0.914·41^-s − 0.482i·43^-s − 0.372i·47^-s + 0.988·49^-s + 0.0777i·53^-s − 0.992·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$900$ = $2^{2} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$53.1017$
Root analytic conductor:	$7.28709$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{900} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 900,\ (\ :3/2),\ 0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.548398105$
$L(\frac12)$	$\approx$	$1.548398105$
$L(\frac{5}{2})$		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 $
	5	$ 1 $
good	7	$ 1 + 2iT - 343T^{2} $
	11	$ 1 + 30T + 1.33e3T^{2} $
	13	$ 1 + 4iT - 2.19e3T^{2} $
	17	$ 1 - 90iT - 4.91e3T^{2} $
	19	$ 1 - 28T + 6.85e3T^{2} $
	23	$ 1 + 120iT - 1.21e4T^{2} $
	29	$ 1 - 210T + 2.43e4T^{2} $
	31	$ 1 + 4T + 2.97e4T^{2} $
	37	$ 1 + 200iT - 5.06e4T^{2} $
	41	$ 1 + 240T + 6.89e4T^{2} $
	43	$ 1 + 136iT - 7.95e4T^{2} $
	47	$ 1 + 120iT - 1.03e5T^{2} $
	53	$ 1 - 30iT - 1.48e5T^{2} $
	59	$ 1 + 450T + 2.05e5T^{2} $
	61	$ 1 + 166T + 2.26e5T^{2} $
	67	$ 1 + 908iT - 3.00e5T^{2} $
	71	$ 1 - 1.02e3T + 3.57e5T^{2} $
	73	$ 1 + 250iT - 3.89e5T^{2} $
	79	$ 1 - 916T + 4.93e5T^{2} $
	83	$ 1 - 1.14e3iT - 5.71e5T^{2} $
	89	$ 1 + 420T + 7.04e5T^{2} $
	97	$ 1 + 1.53e3iT - 9.12e5T^{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

−9.669766547832549444602307137189, −8.560655716411356922320459271382, −8.045841514149026019984096396970, −6.99478329200163005348377374548, −6.12306024057428451509262079667, −5.17244807287499468475779863356, −4.20270811482181738308586973216, −3.08247278493990379848327026725, −1.94750423049145117134403054181, −0.47272824774670665840605765436, 0.984316828859506754651978103772, 2.45856659559515618628989173959, 3.35207164519829761499528186795, 4.72278427671724149084418650943, 5.36829310564883795972252311327, 6.48105692284846780380498347276, 7.39385929967941870811973318823, 8.118443000210158932411939371525, 9.108312963500059114962368417645, 9.859958741930050112163421641324

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.548398105\)
\(L(\frac12)\)	\(\approx\)	\(1.548398105\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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