LMFDB - L-function 2-6030-201.200-c1-0-25

Dirichlet series

L(s) = 1

− 2^-s + 4^-s − 5^-s − 3.48i·7^-s − 8^-s + 10^-s + 3.99·11^-s + 4.23i·13^-s + 3.48i·14^-s + 16^-s − 2.08i·17^-s + 3.38·19^-s − 20^-s − 3.99·22^-s + 3.40i·23^-s + ⋯

L(s) = 1

− 0.707·2^-s + 0.5·4^-s − 0.447·5^-s − 1.31i·7^-s − 0.353·8^-s + 0.316·10^-s + 1.20·11^-s + 1.17i·13^-s + 0.931i·14^-s + 0.250·16^-s − 0.505i·17^-s + 0.776·19^-s − 0.223·20^-s − 0.852·22^-s + 0.709i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6030$ = $2 \cdot 3^{2} \cdot 5 \cdot 67$
Sign:	$0.830 - 0.557i$
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:	$(2,\ 6030,\ (\ :1/2),\ 0.830 - 0.557i)$

Particular Values

$L(1)$	$\approx$	$1.271110959$
$L(\frac12)$	$\approx$	$1.271110959$
$L(\frac{3}{2})$		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 + (7.64 + 2.91i)T $
good	7	$ 1 + 3.48iT - 7T^{2} $
	11	$ 1 - 3.99T + 11T^{2} $
	13	$ 1 - 4.23iT - 13T^{2} $
	17	$ 1 + 2.08iT - 17T^{2} $
	19	$ 1 - 3.38T + 19T^{2} $
	23	$ 1 - 3.40iT - 23T^{2} $
	29	$ 1 - 7.87iT - 29T^{2} $
	31	$ 1 + 4.40iT - 31T^{2} $
	37	$ 1 - 6.85T + 37T^{2} $
	41	$ 1 - 3.37T + 41T^{2} $
	43	$ 1 - 4.24iT - 43T^{2} $
	47	$ 1 - 0.604iT - 47T^{2} $
	53	$ 1 + 12.4T + 53T^{2} $
	59	$ 1 - 6.23iT - 59T^{2} $
	61	$ 1 + 2.59iT - 61T^{2} $
	71	$ 1 - 8.66iT - 71T^{2} $
	73	$ 1 - 10.3T + 73T^{2} $
	79	$ 1 - 0.716iT - 79T^{2} $
	83	$ 1 - 14.3iT - 83T^{2} $
	89	$ 1 - 18.2iT - 89T^{2} $
	97	$ 1 + 4.52iT - 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

−7.921687118277808296985403559651, −7.54753505067312194944285227444, −6.78841697293104352204262336767, −6.46200198503242092187742728837, −5.24657442700993188944449966886, −4.25641157410352105517813369861, −3.83601596963440342044834984701, −2.88400061332016127871907607024, −1.51968289779999955649316626080, −0.943992571710179381918076159978, 0.52555855317182416759517937674, 1.65126474022432338073963076442, 2.66519816965264762954311083310, 3.34051459865146216941360043437, 4.33369909714567807589887967049, 5.29644282824375196532072005897, 6.08765260454848114796551432116, 6.47583588667429281333036635935, 7.60614327608498617617671598988, 8.010091183380516267663825866513

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.271110959\)
\(L(\frac12)\)	\(\approx\)	\(1.271110959\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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