LMFDB - L-function 2-45e2-5.4-c1-0-37

Dirichlet series

L(s) = 1

− i·2^-s + 4^-s + 3i·7^-s − 3i·8^-s − 2·11^-s − 2i·13^-s + 3·14^-s − 16^-s − 4i·17^-s + 8·19^-s + 2i·22^-s + 3i·23^-s − 2·26^-s + 3i·28^-s + 29^-s + ⋯

L(s) = 1

− 0.707i·2^-s + 0.5·4^-s + 1.13i·7^-s − 1.06i·8^-s − 0.603·11^-s − 0.554i·13^-s + 0.801·14^-s − 0.250·16^-s − 0.970i·17^-s + 1.83·19^-s + 0.426i·22^-s + 0.625i·23^-s − 0.392·26^-s + 0.566i·28^-s + 0.185·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2025$ = $3^{4} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$16.1697$
Root analytic conductor:	$4.02115$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2025} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2025,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$2.141060383$
$L(\frac12)$	$\approx$	$2.141060383$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
good	2	$ 1 + iT - 2T^{2} $
	7	$ 1 - 3iT - 7T^{2} $
	11	$ 1 + 2T + 11T^{2} $
	13	$ 1 + 2iT - 13T^{2} $
	17	$ 1 + 4iT - 17T^{2} $
	19	$ 1 - 8T + 19T^{2} $
	23	$ 1 - 3iT - 23T^{2} $
	29	$ 1 - T + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 4iT - 37T^{2} $
	41	$ 1 - 5T + 41T^{2} $
	43	$ 1 + 8iT - 43T^{2} $
	47	$ 1 + 7iT - 47T^{2} $
	53	$ 1 + 2iT - 53T^{2} $
	59	$ 1 - 14T + 59T^{2} $
	61	$ 1 - 7T + 61T^{2} $
	67	$ 1 - 3iT - 67T^{2} $
	71	$ 1 - 2T + 71T^{2} $
	73	$ 1 - 4iT - 73T^{2} $
	79	$ 1 - 6T + 79T^{2} $
	83	$ 1 - 9iT - 83T^{2} $
	89	$ 1 - 15T + 89T^{2} $
	97	$ 1 + 2iT - 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

−9.262100517473119846947611803145, −8.246952852955722579449137054363, −7.43630468478314830095915477430, −6.74887922406124275868822580062, −5.48384601331890464584641640781, −5.30118154035997702690623498826, −3.70198749896840164476878621518, −2.87682520648639803868689590344, −2.26407020116329291878464938219, −0.898631429461865569707751739729, 1.12932340481397501055521220643, 2.42156515848049926800147770809, 3.51554734338535868413404096591, 4.52654511275193127997975982567, 5.43454586698286757171213198556, 6.25428291419858248444103050468, 7.01427774480794209098530282911, 7.64369990263612032540219959141, 8.142337194185448615514503784285, 9.216671908468601068603865209808

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.141060383\)
\(L(\frac12)\)	\(\approx\)	\(2.141060383\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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