LMFDB - L-function 2-675-5.4-c3-0-35

Dirichlet series

L(s) = 1

− 2i·2^-s + 4·4^-s − 24i·8^-s + 10·11^-s + 80i·13^-s − 16·16^-s + 7i·17^-s + 113·19^-s − 20i·22^-s + 81i·23^-s + 160·26^-s + 220·29^-s − 189·31^-s − 160i·32^-s + 14·34^-s + ⋯

L(s) = 1

− 0.707i·2^-s + 0.5·4^-s − 1.06i·8^-s + 0.274·11^-s + 1.70i·13^-s − 0.250·16^-s + 0.0998i·17^-s + 1.36·19^-s − 0.193i·22^-s + 0.734i·23^-s + 1.20·26^-s + 1.40·29^-s − 1.09·31^-s − 0.883i·32^-s + 0.0706·34^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$39.8262$
Root analytic conductor:	$6.31080$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :3/2),\ 0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$2.575760829$
$L(\frac12)$	$\approx$	$2.575760829$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
good	2	$ 1 + 2iT - 8T^{2} $
	7	$ 1 - 343T^{2} $
	11	$ 1 - 10T + 1.33e3T^{2} $
	13	$ 1 - 80iT - 2.19e3T^{2} $
	17	$ 1 - 7iT - 4.91e3T^{2} $
	19	$ 1 - 113T + 6.85e3T^{2} $
	23	$ 1 - 81iT - 1.21e4T^{2} $
	29	$ 1 - 220T + 2.43e4T^{2} $
	31	$ 1 + 189T + 2.97e4T^{2} $
	37	$ 1 - 170iT - 5.06e4T^{2} $
	41	$ 1 + 130T + 6.89e4T^{2} $
	43	$ 1 + 10iT - 7.95e4T^{2} $
	47	$ 1 - 160iT - 1.03e5T^{2} $
	53	$ 1 + 631iT - 1.48e5T^{2} $
	59	$ 1 - 560T + 2.05e5T^{2} $
	61	$ 1 - 229T + 2.26e5T^{2} $
	67	$ 1 - 750iT - 3.00e5T^{2} $
	71	$ 1 - 890T + 3.57e5T^{2} $
	73	$ 1 - 890iT - 3.89e5T^{2} $
	79	$ 1 - 27T + 4.93e5T^{2} $
	83	$ 1 + 429iT - 5.71e5T^{2} $
	89	$ 1 - 750T + 7.04e5T^{2} $
	97	$ 1 + 1.48e3iT - 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

−9.974285234857571612026666419788, −9.496915678336895127381620007259, −8.436911828076371933487949925568, −7.14244796215259338992014450778, −6.72116379195482115827311202346, −5.48339275306930437204098102593, −4.20424765593468196328515851235, −3.29326343535137960601161293335, −2.08738125990028598854785370066, −1.11159248792518270497332993930, 0.849938154992172325223707643723, 2.45572955818027593524656405441, 3.45305088702576324831014540558, 5.05182237100380951611648609313, 5.69376859870243811631358552347, 6.66774978043963571993070967693, 7.54038180731919119104274126039, 8.155621266457682224941099054475, 9.144316958133109146501748389677, 10.31100324848719883772596310311

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.575760829\)
\(L(\frac12)\)	\(\approx\)	\(2.575760829\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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