LMFDB - L-function 2-192-8.5-c3-0-3

Dirichlet series

L(s) = 1

− 3i·3^-s + 10.3i·5^-s + 3.46·7^-s − 9·9^-s + 55.4i·13^-s + 31.1·15^-s − 90·17^-s + 116i·19^-s − 10.3i·21^-s + 103.·23^-s + 17·25^-s + 27i·27^-s + 259. i·29^-s + 301.·31^-s + 36i·35^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.929i·5^-s + 0.187·7^-s − 0.333·9^-s + 1.18i·13^-s + 0.536·15^-s − 1.28·17^-s + 1.40i·19^-s − 0.107i·21^-s + 0.942·23^-s + 0.136·25^-s + 0.192i·27^-s + 1.66i·29^-s + 1.74·31^-s + 0.173i·35^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$192$ = $2^{6} \cdot 3$
Sign:	$0.258 - 0.965i$
Analytic conductor:	$11.3283$
Root analytic conductor:	$3.36576$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{192} (97, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 192,\ (\ :3/2),\ 0.258 - 0.965i)$

Particular Values

$L(2)$	$\approx$	$1.11368 + 0.854559i$
$L(\frac12)$	$\approx$	$1.11368 + 0.854559i$
$L(\frac{5}{2})$		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 - 10.3iT - 125T^{2} $
	7	$ 1 - 3.46T + 343T^{2} $
	11	$ 1 - 1.33e3T^{2} $
	13	$ 1 - 55.4iT - 2.19e3T^{2} $
	17	$ 1 + 90T + 4.91e3T^{2} $
	19	$ 1 - 116iT - 6.85e3T^{2} $
	23	$ 1 - 103.T + 1.21e4T^{2} $
	29	$ 1 - 259. iT - 2.43e4T^{2} $
	31	$ 1 - 301.T + 2.97e4T^{2} $
	37	$ 1 - 34.6iT - 5.06e4T^{2} $
	41	$ 1 + 54T + 6.89e4T^{2} $
	43	$ 1 + 20iT - 7.95e4T^{2} $
	47	$ 1 + 394.T + 1.03e5T^{2} $
	53	$ 1 - 488. iT - 1.48e5T^{2} $
	59	$ 1 + 324iT - 2.05e5T^{2} $
	61	$ 1 + 575. iT - 2.26e5T^{2} $
	67	$ 1 + 116iT - 3.00e5T^{2} $
	71	$ 1 + 1.10e3T + 3.57e5T^{2} $
	73	$ 1 - 1.10e3T + 3.89e5T^{2} $
	79	$ 1 - 148.T + 4.93e5T^{2} $
	83	$ 1 + 1.15e3iT - 5.71e5T^{2} $
	89	$ 1 - 918T + 7.04e5T^{2} $
	97	$ 1 - 190T + 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

−12.21516992556779888799424565687, −11.30049800025477708178733277909, −10.53085923004328071077976770390, −9.227724822903601550069333941686, −8.163115666454852848744559814329, −6.91009598775252913242746513116, −6.39836567773377264526503343090, −4.71862083520080706243362678460, −3.14294753958494508186816991050, −1.71713062226514208791846130543, 0.62707656219436239015030994800, 2.74385832664733611989329597194, 4.44957338670827216535203170801, 5.15376237132896869897160849035, 6.57102280490647491079067241780, 8.099070847895454143489168010785, 8.856170098099650617032604101320, 9.825714956966005245400193394728, 10.92551721093803234054174924686, 11.76358435810635047660206708289

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.11368 + 0.854559i\)
\(L(\frac12)\)	\(\approx\)	\(1.11368 + 0.854559i\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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