L-function 1-1480-1480.59-r0-0-0

• Label: 1-1480-1480.59-r0-0-0
• Degree: $1$
• Conductor: $1480$
• Sign: $-0.994 - 0.103i$
• 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

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

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$1480$    =    $2^{3} \cdot 5 \cdot 37$
Sign:	$-0.994 - 0.103i$
Analytic conductor:	$6.87309$
Root analytic conductor:	$6.87309$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1480} (59, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 1480,\ (0:\ ),\ -0.994 - 0.103i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.04852511950 - 0.9335598269i$
$L(1)$	$\approx$	$0.9205732382 - 0.4437150447i$

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	$1 + (0.766 - 0.642i)T$
	7	$1 + (-0.939 - 0.342i)T$
	11	$1 + (0.5 - 0.866i)T$
	13	$1 + (-0.984 + 0.173i)T$
	17	$1 + (-0.984 - 0.173i)T$
	19	$1 + (0.642 + 0.766i)T$
	23	$1 + (0.866 - 0.5i)T$
	29	$1 + (-0.866 - 0.5i)T$
	31	$1 - iT$

Imaginary part of the first few zeros on the critical line

−20.97255318843920281622033153591, −20.014763271271337341362457751701, −19.76722938174489484374326344377, −19.05141217830213470928663128252, −18.02759268133920769364754646744, −17.15656486455556894436885175601, −16.432257196164052684309100532, −15.54321584800096468437935268913, −15.070504984971363862066561891932, −14.45199852046785501204330293938, −13.217209453271007355948238232675, −13.00405541790262528392692279495, −11.84744550611591805909230220259, −10.993904787869632140652166826677, −9.98270292963319684084622151334, −9.3245580613339243096976888281, −9.07937497633324247999746211739, −7.726225041232045225789921411404, −7.13577324818644507824648837893, −6.147776511123227953542280138344, −4.94785578102865279005264318250, −4.4090783369972791386179229719, −3.257945663586868273066808641574, −2.68557318083977921577660006135, −1.679296383834174399914713574498, 0.293830064334812188906965901004, 1.514029777899320661218331802689, 2.58701968804656227311572116708, 3.3424398375696086778527686608, 4.08457599890407846564015794675, 5.37196632883659082591497185703, 6.46802418245671613960690072218, 6.94743652981451738956822950805, 7.75773258317009012349405049463, 8.800932898717844988297539938968, 9.2890611084295939473132700090, 10.13280374969131698995844938232, 11.15320807604878111675023163309, 12.16335276872742538170404798346, 12.68410048804236164722246340287, 13.65303918156446729816080902740, 13.98921207389252279779274402364, 14.93139213790552346873956839858, 15.69930952729202944916476075231, 16.64322657379197457931765778733, 17.216190802455083811909585771415, 18.27149529577529650856406403064, 19.00378781934914614864111666370, 19.46282857121086766105686817205, 20.15166214667910106344095328661

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