LMFDB - L-function 2-2925-5.4-c1-0-58

Dirichlet series

L(s) = 1

− 0.414i·2^-s + 1.82·4^-s + 2.82i·7^-s − 1.58i·8^-s + 2·11^-s + i·13^-s + 1.17·14^-s + 3·16^-s − 7.65i·17^-s + 2.82·19^-s − 0.828i·22^-s − 4i·23^-s + 0.414·26^-s + 5.17i·28^-s + 2·29^-s + ⋯

L(s) = 1

− 0.292i·2^-s + 0.914·4^-s + 1.06i·7^-s − 0.560i·8^-s + 0.603·11^-s + 0.277i·13^-s + 0.313·14^-s + 0.750·16^-s − 1.85i·17^-s + 0.648·19^-s − 0.176i·22^-s − 0.834i·23^-s + 0.0812·26^-s + 0.977i·28^-s + 0.371·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2925$ = $3^{2} \cdot 5^{2} \cdot 13$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$23.3562$
Root analytic conductor:	$4.83282$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2925} (2224, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2925,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$2.567414704$
$L(\frac12)$	$\approx$	$2.567414704$
$L(\frac{3}{2})$		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 $
	13	$ 1 - iT $
good	2	$ 1 + 0.414iT - 2T^{2} $
	7	$ 1 - 2.82iT - 7T^{2} $
	11	$ 1 - 2T + 11T^{2} $
	17	$ 1 + 7.65iT - 17T^{2} $
	19	$ 1 - 2.82T + 19T^{2} $
	23	$ 1 + 4iT - 23T^{2} $
	29	$ 1 - 2T + 29T^{2} $
	31	$ 1 + 1.17T + 31T^{2} $
	37	$ 1 + 7.65iT - 37T^{2} $
	41	$ 1 + 5.17T + 41T^{2} $
	43	$ 1 - 1.65iT - 43T^{2} $
	47	$ 1 - 11.6iT - 47T^{2} $
	53	$ 1 + 2iT - 53T^{2} $
	59	$ 1 - 7.65T + 59T^{2} $
	61	$ 1 - 13.3T + 61T^{2} $
	67	$ 1 - 6.82iT - 67T^{2} $
	71	$ 1 + 2T + 71T^{2} $
	73	$ 1 + 0.343iT - 73T^{2} $
	79	$ 1 - 11.3T + 79T^{2} $
	83	$ 1 - 3.65iT - 83T^{2} $
	89	$ 1 - 14.8T + 89T^{2} $
	97	$ 1 - 3.65iT - 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

−8.878925045255226430507034036808, −7.88448245392699650906278725017, −7.07795745040335573044612733718, −6.52915409288721407317265713176, −5.65457677001859684880475237834, −4.93030489900748785348192935870, −3.75404560435893750640614808880, −2.76138126953161816048022042855, −2.24132920146476728134256335831, −0.943020994419852943220650855574, 1.12940611922741700079296770363, 1.99418483269219665828802028232, 3.39255877958607424364648779652, 3.85300852874425485903479155126, 5.08152149854055898974082981066, 5.91001534573219021824811310562, 6.68724875475662664340033751251, 7.16060157252890108062250857843, 8.027632820303162828625970683374, 8.508388459134827983871735447866

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.567414704\)
\(L(\frac12)\)	\(\approx\)	\(2.567414704\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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