LMFDB - L-function 2-1232-28.27-c1-0-13

Dirichlet series

L(s) = 1

− 3.31·3^-s − 3.99i·5^-s + (−2.08 + 1.62i)7^-s + 7.95·9^-s − i·11^-s + 3.13i·13^-s + 13.2i·15^-s + 5.11i·17^-s + 3.82·19^-s + (6.90 − 5.39i)21^-s − 0.688i·23^-s − 10.9·25^-s − 16.4·27^-s + 3.25·29^-s + 5.28·31^-s + ⋯

L(s) = 1

− 1.91·3^-s − 1.78i·5^-s + (−0.787 + 0.615i)7^-s + 2.65·9^-s − 0.301i·11^-s + 0.870i·13^-s + 3.41i·15^-s + 1.24i·17^-s + 0.877·19^-s + (1.50 − 1.17i)21^-s − 0.143i·23^-s − 2.19·25^-s − 3.15·27^-s + 0.605·29^-s + 0.949·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1232$ = $2^{4} \cdot 7 \cdot 11$
Sign:	$0.615 + 0.787i$
Analytic conductor:	$9.83756$
Root analytic conductor:	$3.13649$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1232} (111, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1232,\ (\ :1/2),\ 0.615 + 0.787i)$

Particular Values

$L(1)$	$\approx$	$0.6776884770$
$L(\frac12)$	$\approx$	$0.6776884770$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (2.08 - 1.62i)T $
	11	$ 1 + iT $
good	3	$ 1 + 3.31T + 3T^{2} $
	5	$ 1 + 3.99iT - 5T^{2} $
	13	$ 1 - 3.13iT - 13T^{2} $
	17	$ 1 - 5.11iT - 17T^{2} $
	19	$ 1 - 3.82T + 19T^{2} $
	23	$ 1 + 0.688iT - 23T^{2} $
	29	$ 1 - 3.25T + 29T^{2} $
	31	$ 1 - 5.28T + 31T^{2} $
	37	$ 1 + 2.44T + 37T^{2} $
	41	$ 1 - 5.93iT - 41T^{2} $
	43	$ 1 + 6.27iT - 43T^{2} $
	47	$ 1 - 2.87T + 47T^{2} $
	53	$ 1 + 3.01T + 53T^{2} $
	59	$ 1 - 9.67T + 59T^{2} $
	61	$ 1 + 9.62iT - 61T^{2} $
	67	$ 1 - 8.08iT - 67T^{2} $
	71	$ 1 - 0.688iT - 71T^{2} $
	73	$ 1 + 15.9iT - 73T^{2} $
	79	$ 1 + 3.52iT - 79T^{2} $
	83	$ 1 - 13.0T + 83T^{2} $
	89	$ 1 - 14.2iT - 89T^{2} $
	97	$ 1 - 1.59iT - 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

−9.673638583192077150696186498955, −8.944401607379578664103841563941, −8.026762284523630660878472168080, −6.72203984074011171304334799558, −6.09147126743930766300735836702, −5.38998579993127369974370610668, −4.74097794765808848381930821389, −3.88057101512001133482671251311, −1.66909970745195450540031554362, −0.61648751279811722484036588368, 0.75715555148008202370928745598, 2.74052189414249944823428233809, 3.73925838352416533023573313915, 4.91891405117007941942647008717, 5.83564311175047286750868092264, 6.52966991829273651678186471925, 7.13442072182812434509178126970, 7.57607167887713097381476719559, 9.620967923613126484353123294170, 10.18331873241958029274308891807

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6776884770\)
\(L(\frac12)\)	\(\approx\)	\(0.6776884770\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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