LMFDB - L-function 2-6030-201.200-c1-0-5

Dirichlet series

L(s) = 1

− 2^-s + 4^-s − 5^-s − 2.79i·7^-s − 8^-s + 10^-s − 6.41·11^-s + 0.0604i·13^-s + 2.79i·14^-s + 16^-s + 2.85i·17^-s − 6.91·19^-s − 20^-s + 6.41·22^-s + 4.79i·23^-s + ⋯

L(s) = 1

− 0.707·2^-s + 0.5·4^-s − 0.447·5^-s − 1.05i·7^-s − 0.353·8^-s + 0.316·10^-s − 1.93·11^-s + 0.0167i·13^-s + 0.747i·14^-s + 0.250·16^-s + 0.692i·17^-s − 1.58·19^-s − 0.223·20^-s + 1.36·22^-s + 1.00i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6030$ = $2 \cdot 3^{2} \cdot 5 \cdot 67$
Sign:	$0.710 - 0.703i$
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:	$(2,\ 6030,\ (\ :1/2),\ 0.710 - 0.703i)$

Particular Values

$L(1)$	$\approx$	$0.3860842656$
$L(\frac12)$	$\approx$	$0.3860842656$
$L(\frac{3}{2})$		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 + (8.06 + 1.41i)T $
good	7	$ 1 + 2.79iT - 7T^{2} $
	11	$ 1 + 6.41T + 11T^{2} $
	13	$ 1 - 0.0604iT - 13T^{2} $
	17	$ 1 - 2.85iT - 17T^{2} $
	19	$ 1 + 6.91T + 19T^{2} $
	23	$ 1 - 4.79iT - 23T^{2} $
	29	$ 1 + 8.56iT - 29T^{2} $
	31	$ 1 + 8.20iT - 31T^{2} $
	37	$ 1 - 4.75T + 37T^{2} $
	41	$ 1 + 6.22T + 41T^{2} $
	43	$ 1 + 4.96iT - 43T^{2} $
	47	$ 1 + 4.52iT - 47T^{2} $
	53	$ 1 + 1.15T + 53T^{2} $
	59	$ 1 - 3.38iT - 59T^{2} $
	61	$ 1 + 3.47iT - 61T^{2} $
	71	$ 1 - 12.0iT - 71T^{2} $
	73	$ 1 + 6.84T + 73T^{2} $
	79	$ 1 - 10.7iT - 79T^{2} $
	83	$ 1 + 7.74iT - 83T^{2} $
	89	$ 1 - 9.32iT - 89T^{2} $
	97	$ 1 + 1.95iT - 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

−7.947222573083005823070837889610, −7.76339195312320957129362673608, −6.98194149819646631784002552923, −6.12989501134300184734902779851, −5.41833496203981577713813543246, −4.34635499374308357597310354079, −3.83750277851210375433418513743, −2.69246107048195931328504536115, −1.95854209251259293704271696994, −0.56029701675993701178381401408, 0.21268276446418683814840168406, 1.76538460597739641975389314603, 2.72075972133728462473681590670, 3.07041160307628884790705474638, 4.58422266171154683672972795339, 5.08735961016989191500933654493, 5.93069850750087572762173970570, 6.69278916543195214812075457326, 7.43391240422988597513882515517, 8.148060302229372832997663403361

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3860842656\)
\(L(\frac12)\)	\(\approx\)	\(0.3860842656\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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