LMFDB - L-function 2-2e7-8.5-c3-0-11

Dirichlet series

L(s) = 1

− 6.32i·3^-s − 17.8i·5^-s − 22.6·7^-s − 13.0·9^-s + 44.2i·11^-s + 17.8i·13^-s − 113.·15^-s + 70·17^-s − 82.2i·19^-s + 143. i·21^-s − 158.·23^-s − 195.·25^-s − 88.5i·27^-s − 125. i·29^-s + 280·33^-s + ⋯

L(s) = 1

− 1.21i·3^-s − 1.59i·5^-s − 1.22·7^-s − 0.481·9^-s + 1.21i·11^-s + 0.381i·13^-s − 1.94·15^-s + 0.998·17^-s − 0.992i·19^-s + 1.48i·21^-s − 1.43·23^-s − 1.56·25^-s − 0.631i·27^-s − 0.801i·29^-s + 1.47·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$128$ = $2^{7}$
Sign:	$-1$
Analytic conductor:	$7.55224$
Root analytic conductor:	$2.74813$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{128} (65, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 128,\ (\ :3/2),\ -1)$

Particular Values

$L(2)$	$\approx$	$-1.03176i$
$L(\frac12)$	$\approx$	$-1.03176i$
$L(\frac{5}{2})$		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 $
good	3	$ 1 + 6.32iT - 27T^{2} $
	5	$ 1 + 17.8iT - 125T^{2} $
	7	$ 1 + 22.6T + 343T^{2} $
	11	$ 1 - 44.2iT - 1.33e3T^{2} $
	13	$ 1 - 17.8iT - 2.19e3T^{2} $
	17	$ 1 - 70T + 4.91e3T^{2} $
	19	$ 1 + 82.2iT - 6.85e3T^{2} $
	23	$ 1 + 158.T + 1.21e4T^{2} $
	29	$ 1 + 125. iT - 2.43e4T^{2} $
	31	$ 1 + 2.97e4T^{2} $
	37	$ 1 + 375. iT - 5.06e4T^{2} $
	41	$ 1 + 182T + 6.89e4T^{2} $
	43	$ 1 + 132. iT - 7.95e4T^{2} $
	47	$ 1 - 316.T + 1.03e5T^{2} $
	53	$ 1 - 125. iT - 1.48e5T^{2} $
	59	$ 1 + 82.2iT - 2.05e5T^{2} $
	61	$ 1 - 232. iT - 2.26e5T^{2} $
	67	$ 1 - 221. iT - 3.00e5T^{2} $
	71	$ 1 + 113.T + 3.57e5T^{2} $
	73	$ 1 - 910T + 3.89e5T^{2} $
	79	$ 1 - 678.T + 4.93e5T^{2} $
	83	$ 1 + 714. iT - 5.71e5T^{2} $
	89	$ 1 + 546T + 7.04e5T^{2} $
	97	$ 1 + 490T + 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

−12.40832890459561949129560088840, −12.03581223370086710500124360580, −9.947518204547707664068174667346, −9.179098137008116716528552416196, −7.912089215880290909214454444923, −6.93904359422186065305981708579, −5.71317941436571232451105364763, −4.22586096781788846847025218758, −2.01542653512916390805880529757, −0.52240111283131630807705731875, 3.15706713652802246004115415058, 3.62644832231848533663856288426, 5.70107624936722437144644396843, 6.61216925073114069213018639094, 8.082688529071269249181523519771, 9.694926080346705034738807088031, 10.20748072813757285961104199978, 10.93403934088402298490618874132, 12.19068465020443144312628544609, 13.67899429290759603254175071055

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(-1.03176i\)
\(L(\frac12)\)	\(\approx\)	\(-1.03176i\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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