L-function 1-1480-1480.867-r0-0-0

• Label: 1-1480-1480.867-r0-0-0
• Degree: $1$
• Conductor: $1480$
• Sign: $-0.965 - 0.259i$
• Analytic cond.: $6.87309$
• Root an. cond.: $6.87309$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.642 − 0.766i)3^-s + (−0.342 + 0.939i)7^-s + (−0.173 + 0.984i)9^-s + (−0.5 + 0.866i)11^-s + (−0.984 + 0.173i)13^-s + (0.984 + 0.173i)17^-s + (−0.766 + 0.642i)19^-s + (0.939 − 0.342i)21^-s + (0.866 − 0.5i)23^-s + (0.866 − 0.5i)27^-s + (−0.5 + 0.866i)29^-s − 31^-s + (0.984 − 0.173i)33^-s + (0.766 + 0.642i)39^-s + (0.173 + 0.984i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1480$    =    $2^{3} \cdot 5 \cdot 37$
Sign:	$-0.965 - 0.259i$
Analytic conductor:	$6.87309$
Root analytic conductor:	$6.87309$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1480} (867, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1480,\ (0:\ ),\ -0.965 - 0.259i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.001803346862 + 0.01367371111i$
$L(1)$	$\approx$	$0.6360809816 + 0.005869125246i$

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	37	$1$
good	3	$1 + (-0.642 - 0.766i)T$
	7	$1 + (-0.342 + 0.939i)T$
	11	$1 + (-0.5 + 0.866i)T$
	13	$1 + (-0.984 + 0.173i)T$
	17	$1 + (0.984 + 0.173i)T$
	19	$1 + (-0.766 + 0.642i)T$
	23	$1 + (0.866 - 0.5i)T$
	29	$1 + (-0.5 + 0.866i)T$
	31	$1 - T$

Imaginary part of the first few zeros on the critical line

−20.38162228915549847982918294294, −19.49498054275428226974337150649, −18.86426631205967905372372776166, −17.74781646875963579175362924046, −17.0667914237049449639305655937, −16.60526717164078015749192338624, −15.89136523109394184977522051966, −15.00046024442155874694569421190, −14.3589735172663453675041645741, −13.30369696888448091397093575495, −12.696898996953361939704402931093, −11.620067121338353933493435532740, −10.99239860058851598773198203469, −10.270572156443632278151203192429, −9.66615024571286355119549474145, −8.792768743850055571400220191205, −7.6293132680899901456248481829, −6.963783504210203346528052965627, −5.89236438045596834065384875404, −5.24447801256319770276772698828, −4.348716891300456432244444044263, −3.51364221656086796865596528198, −2.7038277468114946176563016951, −1.02545171110361110188022981791, −0.0063704420964365463123180568, 1.636624486434251565045587561326, 2.28067769122951466579069607259, 3.29071949461946493183000930268, 4.77929747731488450062258956810, 5.30514770019903503866695576427, 6.17487961500400491135368922197, 7.00929369639632090356555053576, 7.69790714489716460830924391918, 8.57636680551127522440834284980, 9.60934563560129520640585560962, 10.33255054903430616042635659839, 11.2586380719088890362610553655, 12.15112865518868979914256266029, 12.67037256399706843298380434614, 13.026693700403614611588749558863, 14.55746285181581989640103220110, 14.7596468561677917000625706934, 15.9666210339699618821483068590, 16.67891394296011095169748258542, 17.31842477096708300928404011004, 18.20517608091882328355919780619, 18.76124738292038980642490743043, 19.29982044152223834533438499859, 20.22560405026550161174626605302, 21.20346471578492084966971855153

Graph of the $Z$-function along the critical line