LMFDB - L-function 2-3840-5.4-c1-0-15

Dirichlet series

L(s) = 1

− i·3^-s + (−1.23 − 1.86i)5^-s + 0.746i·7^-s − 9^-s − 5.36·11^-s + 2.92i·13^-s + (−1.86 + 1.23i)15^-s − 2.13i·17^-s − 1.73·19^-s + 0.746·21^-s − 7.49i·23^-s + (−1.92 + 4.61i)25^-s + i·27^-s − 6.74·29^-s + 2.64·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + (−0.554 − 0.832i)5^-s + 0.282i·7^-s − 0.333·9^-s − 1.61·11^-s + 0.811i·13^-s + (−0.480 + 0.320i)15^-s − 0.517i·17^-s − 0.397·19^-s + 0.162·21^-s − 1.56i·23^-s + (−0.385 + 0.922i)25^-s + 0.192i·27^-s − 1.25·29^-s + 0.475·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3840$ = $2^{8} \cdot 3 \cdot 5$
Sign:	$0.832 - 0.554i$
Analytic conductor:	$30.6625$
Root analytic conductor:	$5.53737$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3840} (769, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3840,\ (\ :1/2),\ 0.832 - 0.554i)$

Particular Values

$L(1)$	$\approx$	$0.7576667350$
$L(\frac12)$	$\approx$	$0.7576667350$
$L(\frac{3}{2})$		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 $
	3	$ 1 + iT $
	5	$ 1 + (1.23 + 1.86i)T $
good	7	$ 1 - 0.746iT - 7T^{2} $
	11	$ 1 + 5.36T + 11T^{2} $
	13	$ 1 - 2.92iT - 13T^{2} $
	17	$ 1 + 2.13iT - 17T^{2} $
	19	$ 1 + 1.73T + 19T^{2} $
	23	$ 1 + 7.49iT - 23T^{2} $
	29	$ 1 + 6.74T + 29T^{2} $
	31	$ 1 - 2.64T + 31T^{2} $
	37	$ 1 + 1.07iT - 37T^{2} $
	41	$ 1 - 11.2T + 41T^{2} $
	43	$ 1 - 7.44iT - 43T^{2} $
	47	$ 1 + 1.73iT - 47T^{2} $
	53	$ 1 - 7.72iT - 53T^{2} $
	59	$ 1 - 6.85T + 59T^{2} $
	61	$ 1 + 6.45T + 61T^{2} $
	67	$ 1 - 7.44iT - 67T^{2} $
	71	$ 1 + 13.2T + 71T^{2} $
	73	$ 1 - 0.690iT - 73T^{2} $
	79	$ 1 + 2.64T + 79T^{2} $
	83	$ 1 - 5.85iT - 83T^{2} $
	89	$ 1 - 7.59T + 89T^{2} $
	97	$ 1 - 14.1iT - 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

−8.615107984365353114436850423729, −7.66652295988479473053958081456, −7.41686415714967218457151734170, −6.28469085038232041930772499642, −5.58149943972875765664942948110, −4.75400162080574871032001778779, −4.14901984723255189283719366655, −2.81575045236762805063757112716, −2.16096308730562052727351758028, −0.810927044075421969783626520962, 0.28848559773233745626890572007, 2.13728478573134241164212952601, 3.06357145971603216563451169623, 3.67250001993893019332596974829, 4.54782552248924971595180370982, 5.53091059418141985068830459922, 5.97074567836927382254440442367, 7.24905947634883320230566025192, 7.63640285722011628603003063489, 8.248264234846778604550097012023

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7576667350\)
\(L(\frac12)\)	\(\approx\)	\(0.7576667350\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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