L-function 1-1520-1520.1219-r0-0-0

• Label: 1-1520-1520.1219-r0-0-0
• Degree: $1$
• Conductor: $1520$
• Sign: $-0.608 - 0.793i$
• Analytic cond.: $7.05885$
• Root an. cond.: $7.05885$
• 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 + (0.5 − 0.866i)7^-s + (−0.173 + 0.984i)9^-s + (0.866 − 0.5i)11^-s + (−0.642 + 0.766i)13^-s + (−0.173 − 0.984i)17^-s + (−0.984 + 0.173i)21^-s + (0.939 − 0.342i)23^-s + (0.866 − 0.5i)27^-s + (−0.984 − 0.173i)29^-s + (−0.5 + 0.866i)31^-s + (−0.939 − 0.342i)33^-s − i·37^-s + 39^-s + (0.766 − 0.642i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1520$    =    $2^{4} \cdot 5 \cdot 19$
Sign:	$-0.608 - 0.793i$
Analytic conductor:	$7.05885$
Root analytic conductor:	$7.05885$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1520} (1219, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1520,\ (0:\ ),\ -0.608 - 0.793i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.4932367457 - 0.9993628553i$
$L(1)$	$\approx$	$0.8144624232 - 0.3897838697i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−20.99350478051119190652137250785, −20.233140024879926153517891906091, −19.42479019466058245867476358250, −18.487685856502985610521181496290, −17.649620714591327677973426276245, −17.170973347407330528974004191646, −16.50925778180110411961572477569, −15.265066882624968886472590876846, −15.10579955664929099865798594358, −14.47331001727470214310660262631, −13.03394499410899144511409604559, −12.40920490454293205168702193178, −11.60576056009308433369220123469, −11.05324118768598034706131206808, −10.10489032399700660591612334053, −9.38137799842429446632833241195, −8.72044950299954078513377558864, −7.68996847196477916568035387089, −6.658132996504412197306878402107, −5.77614250933124068541169711374, −5.172358084298032401213602343916, −4.318898820695111514595483959977, −3.45930856337848328996481187441, −2.327888359328983626091101934, −1.200594159274192172050846465247, 0.50719657293311034055652175809, 1.44483992777328841422197203080, 2.36232234736144817493853912906, 3.66019119059126914199848941675, 4.63557088967557371801158041277, 5.32563077442436982080952941122, 6.42483037689576502294569840647, 7.11413437107265899353369812531, 7.5719124856473357058723523824, 8.74405761263748612364932414087, 9.49238726673612693240691316214, 10.70399038740891042284242272214, 11.213499472734777791217345159007, 11.87473506222320007119882243921, 12.68314168217954402338519967427, 13.607489788321370342168326139387, 14.131301576542124576995718413532, 14.82005461172103396254223982170, 16.24183816477816901999192035915, 16.67494088781602007742867044641, 17.305879857048232473638059224106, 18.02814035439195201148282139804, 18.82576306389111005003858351575, 19.528577269528604696162224915024, 20.14512573898411497585883424430

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