L-function 1-115-115.48-r1-0-0

• Label: 1-115-115.48-r1-0-0
• Degree: $1$
• Conductor: $115$
• Sign: $-0.996 + 0.0845i$
• Analytic cond.: $12.3584$
• Root an. cond.: $12.3584$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.909 − 0.415i)2^-s + (−0.281 − 0.959i)3^-s + (0.654 − 0.755i)4^-s + (−0.654 − 0.755i)6^-s + (−0.989 + 0.142i)7^-s + (0.281 − 0.959i)8^-s + (−0.841 + 0.540i)9^-s + (0.415 − 0.909i)11^-s + (−0.909 − 0.415i)12^-s + (−0.989 − 0.142i)13^-s + (−0.841 + 0.540i)14^-s + (−0.142 − 0.989i)16^-s + (−0.755 + 0.654i)17^-s + (−0.540 + 0.841i)18^-s + (0.654 − 0.755i)19^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0845i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$115$    =    $5 \cdot 23$
Sign:	$-0.996 + 0.0845i$
Analytic conductor:	$12.3584$
Root analytic conductor:	$12.3584$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{115} (48, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 115,\ (1:\ ),\ -0.996 + 0.0845i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.07109228344 - 1.677856383i$
$L(1)$	$\approx$	$0.9311879202 - 0.9229494465i$

Euler product

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

	$p$	$F_p(T)$
bad	5	$1$
	23	$1$
good	2	$1 + (0.909 - 0.415i)T$
	3	$1 + (-0.281 - 0.959i)T$
	7	$1 + (-0.989 + 0.142i)T$
	11	$1 + (0.415 - 0.909i)T$
	13	$1 + (-0.989 - 0.142i)T$
	17	$1 + (-0.755 + 0.654i)T$
	19	$1 + (0.654 - 0.755i)T$
	29	$1 + (0.654 + 0.755i)T$
	31	$1 + (-0.959 - 0.281i)T$

Imaginary part of the first few zeros on the critical line

−29.36425519973831640337082809744, −28.861707048993244129026515399322, −27.33268253590942522451372141906, −26.380639912370305333177258636545, −25.49715591708211938629846284644, −24.44631180050633292633836060906, −22.92473395482703045420983683825, −22.60838663264258432230819234538, −21.672378763999730066196881376977, −20.47023915451802958165308979785, −19.72112159833573661281953019716, −17.67367088068933239011025442183, −16.68835546664336944819599101049, −15.87692770607625431183554879458, −14.94956904649015088136131381572, −13.93386754552253295421343643785, −12.523335892085287530400084495650, −11.68411752322292677837647503666, −10.20598420840010938857169943366, −9.2006933180062952293800223946, −7.3518688345690150140842819710, −6.25712797724178380137060530066, −4.96578853601680047067123715615, −3.98223503979501127304532228317, −2.70627280273594073154454121643, 0.50571762023202468569608423831, 2.24092777138098893162271281768, 3.44116626244563594729154666911, 5.24747293871594010283543915442, 6.32245299582408540370077758433, 7.20304038172538642384210211338, 9.05105612696561701785359138981, 10.59633569866255802001989852297, 11.704804236947307109647455630184, 12.65013196078732911659984481547, 13.42338386466081512889900238575, 14.45868427876864109149241528181, 15.84204926078812993392710566943, 16.955137708274637112591005951146, 18.412471586709984214270520150579, 19.59279204503663133384034761288, 19.83740880491786774593009096038, 21.81506644338489528988924796537, 22.25365808152142611269426962316, 23.39621895353033191597499959754, 24.33336933084853235211982832042, 24.98556999238115666460800273701, 26.30249275995827202816446796647, 27.95103562233025577099509840379, 29.06202413438570368690218199926

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