LMFDB - L-function 2-1134-21.20-c1-0-6

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s − 3.89·5^-s + (−2.44 − 1.01i)7^-s + i·8^-s + 3.89i·10^-s + 3.94i·11^-s − 2.84i·13^-s + (−1.01 + 2.44i)14^-s + 16^-s + 0.742·17^-s − 1.78i·19^-s + 3.89·20^-s + 3.94·22^-s + 6.25i·23^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 1.74·5^-s + (−0.923 − 0.383i)7^-s + 0.353i·8^-s + 1.23i·10^-s + 1.18i·11^-s − 0.790i·13^-s + (−0.270 + 0.653i)14^-s + 0.250·16^-s + 0.179·17^-s − 0.409i·19^-s + 0.870·20^-s + 0.840·22^-s + 1.30i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1134$ = $2 \cdot 3^{4} \cdot 7$
Sign:	$0.923 + 0.383i$
Analytic conductor:	$9.05503$
Root analytic conductor:	$3.00915$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1134} (1133, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1134,\ (\ :1/2),\ 0.923 + 0.383i)$

Particular Values

$L(1)$	$\approx$	$0.7310536172$
$L(\frac12)$	$\approx$	$0.7310536172$
$L(\frac{3}{2})$		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 + iT $
	3	$ 1 $
	7	$ 1 + (2.44 + 1.01i)T $
good	5	$ 1 + 3.89T + 5T^{2} $
	11	$ 1 - 3.94iT - 11T^{2} $
	13	$ 1 + 2.84iT - 13T^{2} $
	17	$ 1 - 0.742T + 17T^{2} $
	19	$ 1 + 1.78iT - 19T^{2} $
	23	$ 1 - 6.25iT - 23T^{2} $
	29	$ 1 + 2.88iT - 29T^{2} $
	31	$ 1 + 3.51iT - 31T^{2} $
	37	$ 1 - 3.00T + 37T^{2} $
	41	$ 1 - 10.4T + 41T^{2} $
	43	$ 1 + 0.943T + 43T^{2} $
	47	$ 1 + 2.18T + 47T^{2} $
	53	$ 1 - 53T^{2} $
	59	$ 1 + 0.0211T + 59T^{2} $
	61	$ 1 + 2.46iT - 61T^{2} $
	67	$ 1 - 13.4T + 67T^{2} $
	71	$ 1 + 1.94iT - 71T^{2} $
	73	$ 1 - 4.85iT - 73T^{2} $
	79	$ 1 - 3.63T + 79T^{2} $
	83	$ 1 - 8.05T + 83T^{2} $
	89	$ 1 + 9.26T + 89T^{2} $
	97	$ 1 - 18.8iT - 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

−9.792243329817344268244899650112, −9.180440692739627339612416607081, −7.83802408673925230090370888616, −7.64263640869326638739390717582, −6.58952513956852966319869214980, −5.21332087494766624254764466332, −4.16648376197887970090897323770, −3.64589110023573876381758518461, −2.62659358848284471100686893243, −0.74258482421026082133694681861, 0.53630600132686923691407242329, 2.96937240127870991198472265034, 3.77623630704208065776299625920, 4.58173264424527435358197934471, 5.81405294994418656407961315848, 6.63128626930817963234885360267, 7.33756361680571229882046881002, 8.346234809581838684074658050961, 8.665658297033142026618908720091, 9.649532731475814421756698003054

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7310536172\)
\(L(\frac12)\)	\(\approx\)	\(0.7310536172\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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