LMFDB - L-function 2-96-8.5-c3-0-3

Dirichlet series

L(s) = 1

− 3i·3^-s + 0.612i·5^-s + 22.7·7^-s − 9·9^-s − 60.2i·11^-s − 52.9i·13^-s + 1.83·15^-s + 47.1·17^-s + 29.1i·19^-s − 68.2i·21^-s − 109.·23^-s + 124.·25^-s + 27i·27^-s + 10.4i·29^-s + 220.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.0547i·5^-s + 1.22·7^-s − 0.333·9^-s − 1.65i·11^-s − 1.12i·13^-s + 0.0316·15^-s + 0.672·17^-s + 0.352i·19^-s − 0.709i·21^-s − 0.992·23^-s + 0.996·25^-s + 0.192i·27^-s + 0.0667i·29^-s + 1.27·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.903i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$96$ = $2^{5} \cdot 3$
Sign:	$0.427 + 0.903i$
Analytic conductor:	$5.66418$
Root analytic conductor:	$2.37995$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{96} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 96,\ (\ :3/2),\ 0.427 + 0.903i)$

Particular Values

$L(2)$	$\approx$	$1.40024 - 0.886509i$
$L(\frac12)$	$\approx$	$1.40024 - 0.886509i$
$L(\frac{5}{2})$		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 + 3iT $
good	5	$ 1 - 0.612iT - 125T^{2} $
	7	$ 1 - 22.7T + 343T^{2} $
	11	$ 1 + 60.2iT - 1.33e3T^{2} $
	13	$ 1 + 52.9iT - 2.19e3T^{2} $
	17	$ 1 - 47.1T + 4.91e3T^{2} $
	19	$ 1 - 29.1iT - 6.85e3T^{2} $
	23	$ 1 + 109.T + 1.21e4T^{2} $
	29	$ 1 - 10.4iT - 2.43e4T^{2} $
	31	$ 1 - 220.T + 2.97e4T^{2} $
	37	$ 1 - 408. iT - 5.06e4T^{2} $
	41	$ 1 + 360.T + 6.89e4T^{2} $
	43	$ 1 - 236. iT - 7.95e4T^{2} $
	47	$ 1 + 129.T + 1.03e5T^{2} $
	53	$ 1 - 117. iT - 1.48e5T^{2} $
	59	$ 1 - 262. iT - 2.05e5T^{2} $
	61	$ 1 + 273. iT - 2.26e5T^{2} $
	67	$ 1 - 89.4iT - 3.00e5T^{2} $
	71	$ 1 - 350.T + 3.57e5T^{2} $
	73	$ 1 - 532.T + 3.89e5T^{2} $
	79	$ 1 - 166.T + 4.93e5T^{2} $
	83	$ 1 + 361. iT - 5.71e5T^{2} $
	89	$ 1 - 40.3T + 7.04e5T^{2} $
	97	$ 1 + 614.T + 9.12e5T^{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

−13.44311101427324408421837291811, −12.13128679009365501666463664687, −11.27171268945163775357138239647, −10.24235431740687672921695475086, −8.333847521904450220817807060741, −8.037388212965681409055556069259, −6.28294529778078933220065088040, −5.12778777022101366021161096721, −3.12210026545588998535672609522, −1.09999676689116059091841406697, 1.96172237674734729345538275186, 4.22645757138495605512171968302, 5.10358281834891909894618662917, 6.92620319782875688718453191580, 8.163604944758821765782660197407, 9.393208089701774574074209121972, 10.39166112859402680759411472612, 11.57905096378395016030190116622, 12.37128819844717534980187252489, 13.99196963554436405536025241066

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.40024 - 0.886509i\)
\(L(\frac12)\)	\(\approx\)	\(1.40024 - 0.886509i\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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