LMFDB - L-function 2-320-8.5-c3-0-20

Dirichlet series

L(s) = 1

− 7.35i·3^-s − 5i·5^-s + 26.7·7^-s − 27.1·9^-s − 16.7i·11^-s − 83.5i·13^-s − 36.7·15^-s + 102.·17^-s + 83.7i·19^-s − 197. i·21^-s + 73.9·23^-s − 25·25^-s + 1.12i·27^-s + 94.3i·29^-s − 126.·31^-s + ⋯

L(s) = 1

− 1.41i·3^-s − 0.447i·5^-s + 1.44·7^-s − 1.00·9^-s − 0.458i·11^-s − 1.78i·13^-s − 0.633·15^-s + 1.46·17^-s + 1.01i·19^-s − 2.04i·21^-s + 0.670·23^-s − 0.200·25^-s + 0.00804i·27^-s + 0.604i·29^-s − 0.732·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$320$ = $2^{6} \cdot 5$
Sign:	$-0.707 + 0.707i$
Analytic conductor:	$18.8806$
Root analytic conductor:	$4.34518$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{320} (161, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 320,\ (\ :3/2),\ -0.707 + 0.707i)$

Particular Values

$L(2)$	$\approx$	$2.144204172$
$L(\frac12)$	$\approx$	$2.144204172$
$L(\frac{5}{2})$		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 $
	5	$ 1 + 5iT $
good	3	$ 1 + 7.35iT - 27T^{2} $
	7	$ 1 - 26.7T + 343T^{2} $
	11	$ 1 + 16.7iT - 1.33e3T^{2} $
	13	$ 1 + 83.5iT - 2.19e3T^{2} $
	17	$ 1 - 102.T + 4.91e3T^{2} $
	19	$ 1 - 83.7iT - 6.85e3T^{2} $
	23	$ 1 - 73.9T + 1.21e4T^{2} $
	29	$ 1 - 94.3iT - 2.43e4T^{2} $
	31	$ 1 + 126.T + 2.97e4T^{2} $
	37	$ 1 + 210iT - 5.06e4T^{2} $
	41	$ 1 + 334.T + 6.89e4T^{2} $
	43	$ 1 - 297. iT - 7.95e4T^{2} $
	47	$ 1 + 553.T + 1.03e5T^{2} $
	53	$ 1 + 490. iT - 1.48e5T^{2} $
	59	$ 1 - 54iT - 2.05e5T^{2} $
	61	$ 1 - 385. iT - 2.26e5T^{2} $
	67	$ 1 - 1.05e3iT - 3.00e5T^{2} $
	71	$ 1 - 220.T + 3.57e5T^{2} $
	73	$ 1 - 210.T + 3.89e5T^{2} $
	79	$ 1 - 677.T + 4.93e5T^{2} $
	83	$ 1 - 184. iT - 5.71e5T^{2} $
	89	$ 1 - 150.T + 7.04e5T^{2} $
	97	$ 1 - 663.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

−11.07350507723043259530758220484, −10.02938485498838776813782469099, −8.347424108201577679874189470449, −8.080253226742473260343126877902, −7.26333858271316297089828508673, −5.78055507425302685113878939602, −5.16160669463676967579233461582, −3.29324933845407010934328648978, −1.67417688972948652018982633462, −0.854886545591365439832253123225, 1.80188209013079158361016371792, 3.47147138833673080460070995921, 4.61487918653703554039234082644, 5.11953285547526917924754675196, 6.73161236654321694931259550155, 7.84409811824358542014736503744, 8.996344447876482407867759578339, 9.695556981013754992125982269502, 10.66403822076489720577580943707, 11.36885291539714845352652963196

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.144204172\)
\(L(\frac12)\)	\(\approx\)	\(2.144204172\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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