LMFDB - L-function 2-675-15.14-c2-0-25

Dirichlet series

L(s) = 1

− 2.61·2^-s + 2.85·4^-s + 9.70i·7^-s + 3·8^-s − 14.1i·11^-s − 5.70i·13^-s − 25.4i·14^-s − 19.2·16^-s + 5.94·17^-s − 32.4·19^-s + 37.1i·22^-s + 1.58·23^-s + 14.9i·26^-s + 27.7i·28^-s + 2.06i·29^-s + ⋯

L(s) = 1

− 1.30·2^-s + 0.713·4^-s + 1.38i·7^-s + 0.375·8^-s − 1.28i·11^-s − 0.439i·13^-s − 1.81i·14^-s − 1.20·16^-s + 0.349·17^-s − 1.70·19^-s + 1.68i·22^-s + 0.0688·23^-s + 0.574i·26^-s + 0.989i·28^-s + 0.0713i·29^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$18.3924$
Root analytic conductor:	$4.28863$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (674, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :1),\ 0.447 + 0.894i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.5991088700$
$L(\frac12)$	$\approx$	$0.5991088700$
$L(2)$		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 + 2.61T + 4T^{2} $
	7	$ 1 - 9.70iT - 49T^{2} $
	11	$ 1 + 14.1iT - 121T^{2} $
	13	$ 1 + 5.70iT - 169T^{2} $
	17	$ 1 - 5.94T + 289T^{2} $
	19	$ 1 + 32.4T + 361T^{2} $
	23	$ 1 - 1.58T + 529T^{2} $
	29	$ 1 - 2.06iT - 841T^{2} $
	31	$ 1 - 49.2T + 961T^{2} $
	37	$ 1 - 26.5iT - 1.36e3T^{2} $
	41	$ 1 + 65.2iT - 1.68e3T^{2} $
	43	$ 1 - 49.3iT - 1.84e3T^{2} $
	47	$ 1 + 70.1T + 2.20e3T^{2} $
	53	$ 1 - 28.6T + 2.80e3T^{2} $
	59	$ 1 + 95.2iT - 3.48e3T^{2} $
	61	$ 1 - 19T + 3.72e3T^{2} $
	67	$ 1 + 107. iT - 4.48e3T^{2} $
	71	$ 1 - 75.2iT - 5.04e3T^{2} $
	73	$ 1 + 96.7iT - 5.32e3T^{2} $
	79	$ 1 - 101.T + 6.24e3T^{2} $
	83	$ 1 + 44.6T + 6.88e3T^{2} $
	89	$ 1 + 56.2iT - 7.92e3T^{2} $
	97	$ 1 + 132. iT - 9.40e3T^{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

−10.00433890424202073814239283001, −9.130658529747800804693320352545, −8.328422166823463382187578856150, −8.181777945426001665959184805729, −6.63521717735912869021892917894, −5.88636104006275099049156315005, −4.72846113362431309110236555415, −3.11959650746473241301572570864, −1.96296581437555210386049974862, −0.41676208437313576133459916042, 1.00676078411075651260891771537, 2.19188685896074619933147380002, 4.08388660950456453096317425166, 4.67965801789320651249209930795, 6.52196851961997051563699180485, 7.13190193842903140804733951040, 7.921090662276098430886134758913, 8.712507052032461269051856465977, 9.790713831374389220499167452122, 10.19070081322204433565279321297

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 $\F_{q}$
Belyi maps

\(L(\frac{3}{2})\)	\(\approx\)	\(0.5991088700\)
\(L(\frac12)\)	\(\approx\)	\(0.5991088700\)
\(L(2)\)		not available
\(L(1)\)		not available

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