LMFDB - L-function 2-30e2-5.4-c3-0-10

Dirichlet series

L(s) = 1

− 26i·7^-s − 45·11^-s + 44i·13^-s + 117i·17^-s + 91·19^-s + 18i·23^-s + 144·29^-s + 26·31^-s + 214i·37^-s + 459·41^-s − 460i·43^-s − 468i·47^-s − 333·49^-s − 558i·53^-s − 72·59^-s + ⋯

L(s) = 1

− 1.40i·7^-s − 1.23·11^-s + 0.938i·13^-s + 1.66i·17^-s + 1.09·19^-s + 0.163i·23^-s + 0.922·29^-s + 0.150·31^-s + 0.950i·37^-s + 1.74·41^-s − 1.63i·43^-s − 1.45i·47^-s − 0.970·49^-s − 1.44i·53^-s − 0.158·59^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$900$ = $2^{2} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
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:	$(2,\ 900,\ (\ :3/2),\ 0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.807629786$
$L(\frac12)$	$\approx$	$1.807629786$
$L(\frac{5}{2})$		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 + 26iT - 343T^{2} $
	11	$ 1 + 45T + 1.33e3T^{2} $
	13	$ 1 - 44iT - 2.19e3T^{2} $
	17	$ 1 - 117iT - 4.91e3T^{2} $
	19	$ 1 - 91T + 6.85e3T^{2} $
	23	$ 1 - 18iT - 1.21e4T^{2} $
	29	$ 1 - 144T + 2.43e4T^{2} $
	31	$ 1 - 26T + 2.97e4T^{2} $
	37	$ 1 - 214iT - 5.06e4T^{2} $
	41	$ 1 - 459T + 6.89e4T^{2} $
	43	$ 1 + 460iT - 7.95e4T^{2} $
	47	$ 1 + 468iT - 1.03e5T^{2} $
	53	$ 1 + 558iT - 1.48e5T^{2} $
	59	$ 1 + 72T + 2.05e5T^{2} $
	61	$ 1 + 118T + 2.26e5T^{2} $
	67	$ 1 + 251iT - 3.00e5T^{2} $
	71	$ 1 + 108T + 3.57e5T^{2} $
	73	$ 1 - 299iT - 3.89e5T^{2} $
	79	$ 1 - 898T + 4.93e5T^{2} $
	83	$ 1 + 927iT - 5.71e5T^{2} $
	89	$ 1 - 351T + 7.04e5T^{2} $
	97	$ 1 + 386iT - 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

−9.961561863717592613599086285642, −8.726391812419480224316502356541, −7.88082885618974828707357208981, −7.19979820970831366222830228002, −6.30955874008357491673509176423, −5.18211229203950284656880266285, −4.22551167141906833263827476392, −3.37807810286614379029293629640, −1.94142565109168197670730036459, −0.67788964809847291869451400545, 0.78686815368776040641368353560, 2.66984050521196832909132661843, 2.85799980170564249948496543490, 4.72686011285505759799352115041, 5.41064693233824175360565809734, 6.10524315342643758471420295662, 7.51019493538969357268979159879, 7.960445563035771737172922084198, 9.107488787389368882770294826027, 9.599414075431260877219542661491

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.807629786\)
\(L(\frac12)\)	\(\approx\)	\(1.807629786\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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