LMFDB - L-function 2-1932-161.160-c1-0-6

Dirichlet series

L(s) = 1

− i·3^-s − 0.248·5^-s + (−0.652 + 2.56i)7^-s − 9^-s − 4.05i·11^-s + 5.46i·13^-s + 0.248i·15^-s + 0.349·17^-s + 3.87·19^-s + (2.56 + 0.652i)21^-s + (0.300 − 4.78i)23^-s − 4.93·25^-s + i·27^-s − 3.50·29^-s + 7.87i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.110·5^-s + (−0.246 + 0.969i)7^-s − 0.333·9^-s − 1.22i·11^-s + 1.51i·13^-s + 0.0640i·15^-s + 0.0848·17^-s + 0.890·19^-s + (0.559 + 0.142i)21^-s + (0.0626 − 0.998i)23^-s − 0.987·25^-s + 0.192i·27^-s − 0.651·29^-s + 1.41i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1932$ = $2^{2} \cdot 3 \cdot 7 \cdot 23$
Sign:	$0.306 - 0.951i$
Analytic conductor:	$15.4270$
Root analytic conductor:	$3.92773$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1932} (1609, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1932,\ (\ :1/2),\ 0.306 - 0.951i)$

Particular Values

$L(1)$	$\approx$	$1.169311548$
$L(\frac12)$	$\approx$	$1.169311548$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (0.652 - 2.56i)T $
	23	$ 1 + (-0.300 + 4.78i)T $
good	5	$ 1 + 0.248T + 5T^{2} $
	11	$ 1 + 4.05iT - 11T^{2} $
	13	$ 1 - 5.46iT - 13T^{2} $
	17	$ 1 - 0.349T + 17T^{2} $
	19	$ 1 - 3.87T + 19T^{2} $
	29	$ 1 + 3.50T + 29T^{2} $
	31	$ 1 - 7.87iT - 31T^{2} $
	37	$ 1 + 3.76iT - 37T^{2} $
	41	$ 1 - 5.40iT - 41T^{2} $
	43	$ 1 - 11.4iT - 43T^{2} $
	47	$ 1 - 1.86iT - 47T^{2} $
	53	$ 1 - 9.71iT - 53T^{2} $
	59	$ 1 - 12.5iT - 59T^{2} $
	61	$ 1 - 4.03T + 61T^{2} $
	67	$ 1 - 4.70iT - 67T^{2} $
	71	$ 1 - 13.9T + 71T^{2} $
	73	$ 1 - 9.64iT - 73T^{2} $
	79	$ 1 + 6.98iT - 79T^{2} $
	83	$ 1 + 12.2T + 83T^{2} $
	89	$ 1 - 6.84T + 89T^{2} $
	97	$ 1 - 1.02T + 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

−9.120126746550152778549469278955, −8.674410429875716817214154635730, −7.83726324976780752626730694976, −6.91319587353555018349349547228, −6.16558876842961455164283221538, −5.58063097020632175139316103851, −4.46309345107046251778484342198, −3.31979394212391708659985707968, −2.46826991878788997580989118786, −1.28395074251369379460948229118, 0.44504971103888385280509739633, 2.03439847396058273492630299866, 3.43685208943086271960485782226, 3.88630630869087604259427792876, 5.04759395847918786967078619663, 5.60559972848547217110789849348, 6.80570691885732249759387544137, 7.62825433701123880898494005372, 7.998968541518483739637256405772, 9.354993215430554274751879118814

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.169311548\)
\(L(\frac12)\)	\(\approx\)	\(1.169311548\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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