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Properties

Label 1-115-115.27-r1-0-0
Degree $1$
Conductor $115$
Sign $0.783 + 0.622i$
Analytic cond. $12.3584$
Root an. cond. $12.3584$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.755 − 0.654i)2-s + (−0.540 − 0.841i)3-s + (0.142 + 0.989i)4-s + (−0.142 + 0.989i)6-s + (−0.281 − 0.959i)7-s + (0.540 − 0.841i)8-s + (−0.415 + 0.909i)9-s + (−0.654 − 0.755i)11-s + (0.755 − 0.654i)12-s + (−0.281 + 0.959i)13-s + (−0.415 + 0.909i)14-s + (−0.959 + 0.281i)16-s + (−0.989 − 0.142i)17-s + (0.909 − 0.415i)18-s + (0.142 + 0.989i)19-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)2-s + (−0.540 − 0.841i)3-s + (0.142 + 0.989i)4-s + (−0.142 + 0.989i)6-s + (−0.281 − 0.959i)7-s + (0.540 − 0.841i)8-s + (−0.415 + 0.909i)9-s + (−0.654 − 0.755i)11-s + (0.755 − 0.654i)12-s + (−0.281 + 0.959i)13-s + (−0.415 + 0.909i)14-s + (−0.959 + 0.281i)16-s + (−0.989 − 0.142i)17-s + (0.909 − 0.415i)18-s + (0.142 + 0.989i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.783 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.783 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.783 + 0.622i$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ 0.783 + 0.622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2877303627 + 0.1003771155i\)
\(L(\frac12)\) \(\approx\) \(0.2877303627 + 0.1003771155i\)
\(L(1)\) \(\approx\) \(0.4401231465 - 0.2198303921i\)
\(L(1)\) \(\approx\) \(0.4401231465 - 0.2198303921i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.755 - 0.654i)T \)
3 \( 1 + (-0.540 - 0.841i)T \)
7 \( 1 + (-0.281 - 0.959i)T \)
11 \( 1 + (-0.654 - 0.755i)T \)
13 \( 1 + (-0.281 + 0.959i)T \)
17 \( 1 + (-0.989 - 0.142i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.142 - 0.989i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (0.909 + 0.415i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.540 - 0.841i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.281 + 0.959i)T \)
59 \( 1 + (0.959 + 0.281i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
67 \( 1 + (-0.755 - 0.654i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.989 + 0.142i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (-0.909 - 0.415i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.909 + 0.415i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.3612142833952787956136776130, −28.077769771717960143986804767808, −26.8668497354244312250598161983, −26.02026050914091151723795213710, −25.08019158415365286952662375224, −23.91482522237798417458049185066, −22.791922178552876952334499158518, −21.96258132378781117684355356331, −20.56902799435701821428844416738, −19.577822115422299737664852827712, −18.02663544936807611817330216852, −17.65291144421965145053447681547, −16.23454149138319438341985188107, −15.42554187765189865866125093424, −14.89220750136050367242044396631, −12.95514435456427762513667214566, −11.49673958515946661898016030661, −10.38122319883573804892287896630, −9.4719171405136508309323901200, −8.44531205817366461323496277745, −6.883918168029995320557496884747, −5.6500652624001038311094672424, −4.77242973295006414555281133898, −2.56792212375085787093183181594, −0.1986907855378733471847928274, 1.13550753239498224345005898770, 2.626244363644392428152798911875, 4.33133852680107936468942118791, 6.32184992050522127228192271662, 7.38482690823458334850707529943, 8.40050106215665423867449788711, 9.96079923771497997733193619756, 10.99691950900294942195497126218, 11.869563890922168510714084779689, 13.125392708513373466350368867126, 13.83938190058060907328871038904, 16.13408221683858398227791250827, 16.84867105512385397512165940343, 17.83341927109928133197840124074, 18.86641164419890102319436334347, 19.55484856122232160371447255789, 20.72163341267432477349283703150, 21.904249812944755129486025824266, 23.015315807292564732200211100608, 24.06175163751454323352300492699, 25.13588492321811363928068024123, 26.44832296520263656146611620689, 27.00140169777615542220194474366, 28.58334873239130499746370491868, 29.02053634881252608744613694116

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