LMFDB - L-function 2-6030-201.200-c1-0-47

Dirichlet series

L(s) = 1

− 2^-s + 4^-s − 5^-s − 0.376i·7^-s − 8^-s + 10^-s + 3.43·11^-s + 3.04i·13^-s + 0.376i·14^-s + 16^-s − 5.53i·17^-s + 4.42·19^-s − 20^-s − 3.43·22^-s − 2.14i·23^-s + ⋯

L(s) = 1

− 0.707·2^-s + 0.5·4^-s − 0.447·5^-s − 0.142i·7^-s − 0.353·8^-s + 0.316·10^-s + 1.03·11^-s + 0.843i·13^-s + 0.100i·14^-s + 0.250·16^-s − 1.34i·17^-s + 1.01·19^-s − 0.223·20^-s − 0.732·22^-s − 0.446i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6030$ = $2 \cdot 3^{2} \cdot 5 \cdot 67$
Sign:	$0.851 + 0.524i$
Analytic conductor:	$48.1497$
Root analytic conductor:	$6.93900$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6030} (2411, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6030,\ (\ :1/2),\ 0.851 + 0.524i)$

Particular Values

$L(1)$	$\approx$	$1.340918755$
$L(\frac12)$	$\approx$	$1.340918755$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 + T $
	3	$ 1 $
	5	$ 1 + T $
	67	$ 1 + (7.52 - 3.21i)T $
good	7	$ 1 + 0.376iT - 7T^{2} $
	11	$ 1 - 3.43T + 11T^{2} $
	13	$ 1 - 3.04iT - 13T^{2} $
	17	$ 1 + 5.53iT - 17T^{2} $
	19	$ 1 - 4.42T + 19T^{2} $
	23	$ 1 + 2.14iT - 23T^{2} $
	29	$ 1 - 1.98iT - 29T^{2} $
	31	$ 1 + 2.86iT - 31T^{2} $
	37	$ 1 + 0.233T + 37T^{2} $
	41	$ 1 - 1.78T + 41T^{2} $
	43	$ 1 - 9.77iT - 43T^{2} $
	47	$ 1 - 5.43iT - 47T^{2} $
	53	$ 1 - 5.11T + 53T^{2} $
	59	$ 1 + 14.9iT - 59T^{2} $
	61	$ 1 + 11.6iT - 61T^{2} $
	71	$ 1 + 9.67iT - 71T^{2} $
	73	$ 1 + 9.52T + 73T^{2} $
	79	$ 1 - 5.92iT - 79T^{2} $
	83	$ 1 + 5.42iT - 83T^{2} $
	89	$ 1 - 3.00iT - 89T^{2} $
	97	$ 1 - 11.2iT - 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

−7.941320876923277705154314746914, −7.37534837231078324880568485050, −6.76128515485560416392789580856, −6.15495999901556644904844033889, −5.07260581133815785569049415423, −4.35480402758117036191314778204, −3.48246038095189275638957275671, −2.65299166696568579144678559320, −1.53307523962305345393139594519, −0.61330382362001489664179291792, 0.831925452999845742382132231679, 1.68897584199613603488442659869, 2.85030129828917336026422657761, 3.65990347056175488128159336154, 4.31895692936239263074547722650, 5.64653193105323905135367344965, 5.87269198310733716676027793203, 7.15916479101665100716201633817, 7.25054626416310190910397113350, 8.360504164907825069648266902334

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.340918755\)
\(L(\frac12)\)	\(\approx\)	\(1.340918755\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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