LMFDB - L-function 2-1350-5.4-c3-0-26

Dirichlet series

L(s) = 1

− 2i·2^-s − 4·4^-s + 23i·7^-s + 8i·8^-s + 30·11^-s + 34i·13^-s + 46·14^-s + 16·16^-s − 42i·17^-s + 139·19^-s − 60i·22^-s − 192i·23^-s + 68·26^-s − 92i·28^-s − 234·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s + 1.24i·7^-s + 0.353i·8^-s + 0.822·11^-s + 0.725i·13^-s + 0.878·14^-s + 0.250·16^-s − 0.599i·17^-s + 1.67·19^-s − 0.581i·22^-s − 1.74i·23^-s + 0.512·26^-s − 0.620i·28^-s − 1.49·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1350$ = $2 \cdot 3^{3} \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$79.6525$
Root analytic conductor:	$8.92482$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1350} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1350,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.948077068$
$L(\frac12)$	$\approx$	$1.948077068$
$L(\frac{5}{2})$		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 + 2iT $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 - 23iT - 343T^{2} $
	11	$ 1 - 30T + 1.33e3T^{2} $
	13	$ 1 - 34iT - 2.19e3T^{2} $
	17	$ 1 + 42iT - 4.91e3T^{2} $
	19	$ 1 - 139T + 6.85e3T^{2} $
	23	$ 1 + 192iT - 1.21e4T^{2} $
	29	$ 1 + 234T + 2.43e4T^{2} $
	31	$ 1 + 55T + 2.97e4T^{2} $
	37	$ 1 - 191iT - 5.06e4T^{2} $
	41	$ 1 - 138T + 6.89e4T^{2} $
	43	$ 1 + 53iT - 7.95e4T^{2} $
	47	$ 1 - 366iT - 1.03e5T^{2} $
	53	$ 1 - 330iT - 1.48e5T^{2} $
	59	$ 1 - 396T + 2.05e5T^{2} $
	61	$ 1 - 23T + 2.26e5T^{2} $
	67	$ 1 - 452iT - 3.00e5T^{2} $
	71	$ 1 - 204T + 3.57e5T^{2} $
	73	$ 1 - 691iT - 3.89e5T^{2} $
	79	$ 1 - 709T + 4.93e5T^{2} $
	83	$ 1 + 1.09e3iT - 5.71e5T^{2} $
	89	$ 1 - 816T + 7.04e5T^{2} $
	97	$ 1 - 905iT - 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

−9.198332325488581750882798452470, −8.918711807810749363964605666697, −7.80517189333905903915727446340, −6.79520705165701299812025387110, −5.84408649574187243833065410715, −5.03118314197903679268832560075, −4.06120804337144436867189475944, −2.98233355757029309152521709874, −2.14823276783165838146648620826, −0.996711207653808070261584069649, 0.54816388286798262476736766119, 1.56476808537157535037200275645, 3.52749027710271547491704356211, 3.84046592406864886065674703122, 5.19142333856758056245891650680, 5.79654734428314153900682027796, 6.95858885433513371841813469362, 7.45412282331189108906789588007, 8.089315436676468617745924990323, 9.333771564473443982250173851948

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.948077068\)
\(L(\frac12)\)	\(\approx\)	\(1.948077068\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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