L-function 1-1480-1480.147-r0-0-0

• Label: 1-1480-1480.147-r0-0-0
• Degree: $1$
• Conductor: $1480$
• Sign: $-0.850 - 0.525i$
• 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

− i·3^-s − i·7^-s − 9^-s + 11^-s − i·13^-s − i·17^-s + 19^-s − 21^-s − i·23^-s + i·27^-s − 29^-s + 31^-s − i·33^-s − 39^-s + 41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1480$    =    $2^{3} \cdot 5 \cdot 37$
Sign:	$-0.850 - 0.525i$
Analytic conductor:	$6.87309$
Root analytic conductor:	$6.87309$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1480} (147, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1480,\ (0:\ ),\ -0.850 - 0.525i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.4161273951 - 1.464829610i$
$L(1)$	$\approx$	$0.9031452314 - 0.6412356449i$

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	$1 + T$
	7	$1 - iT$
	11	$1$
	13	$1$
	17	$1$
	19	$1 - iT$
	23	$1$
	29	$1 - T$
	31	$1$

Imaginary part of the first few zeros on the critical line

−21.131852503979515065510136184229, −20.33401835886887004785519125881, −19.35640683987228156379279406239, −19.01038227623975324475514910674, −17.73778169040339271495618760550, −17.21602124687175638598928575632, −16.30191287214280469138216216526, −15.759139788875553924650058996417, −14.91036377925254939250153521672, −14.412947764066485693937403163994, −13.557687817315782421341557874422, −12.37630901021512045685507999458, −11.60475828581893328536427706066, −11.23780197604042629012600001932, −9.9982831440603782649271073240, −9.359383608110354963040411170054, −8.87384142982389289317130306783, −7.95772159645824757959975644563, −6.72567704700964140456543718132, −5.87442987018099598797454137905, −5.2275956935351423372934727188, −4.14314028136176539599861704874, −3.57623411938214563695067137242, −2.47580949501013067625561606782, −1.44793663606853065704975310059, 0.641540535522301554039404838474, 1.265664950819015169510394845458, 2.569505594509637274348755125759, 3.371492035733630982685071123461, 4.42985207460449764473382160949, 5.480699863912272524984003202384, 6.36229328751180785633903847193, 7.1725234005608543825851073362, 7.66289119856724833105724416385, 8.59096545611020577260402540031, 9.5226615979570096523259202653, 10.39942240919467652877705081856, 11.37859430419103030237364501832, 11.89614871479613752595773313635, 12.884301554474505245333541071589, 13.45644221085132164971819444667, 14.22973070801913416824123152045, 14.7465064955624874926274212202, 16.035112051907190658723969538982, 16.7145702376677028954230802290, 17.51061691462599265980480116819, 17.99776191662624484327956226007, 18.87406518245815372685538039150, 19.6473266470308427147394076373, 20.32591123182763270841016501368

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