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Properties

Label 1-115-115.104-r0-0-0
Degree $1$
Conductor $115$
Sign $0.451 + 0.892i$
Analytic cond. $0.534057$
Root an. cond. $0.534057$
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  + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.654 + 0.755i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.415 − 0.909i)12-s + (0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (0.654 − 0.755i)17-s + (−0.841 + 0.540i)18-s + (−0.654 − 0.755i)19-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (0.959 + 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.654 + 0.755i)6-s + (0.142 − 0.989i)7-s + (0.959 − 0.281i)8-s + (0.841 + 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.415 − 0.909i)12-s + (0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (0.654 − 0.755i)17-s + (−0.841 + 0.540i)18-s + (−0.654 − 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.451 + 0.892i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ 0.451 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9653060489 + 0.5931053764i\)
\(L(\frac12)\) \(\approx\) \(0.9653060489 + 0.5931053764i\)
\(L(1)\) \(\approx\) \(1.017174988 + 0.4563193137i\)
\(L(1)\) \(\approx\) \(1.017174988 + 0.4563193137i\)

Euler product

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

−29.36168942459834257453357744674, −27.89522671498788679021518349605, −27.32964855871603707270038921905, −26.07039660679023265196630762666, −25.28995058231709820058737083207, −24.27196549210104892262650497322, −22.6857169719161010775486708084, −21.50671504891928325535223493490, −20.888972981284871630264652893789, −19.66725898635873456311540660030, −18.92182229145737689738769900442, −18.17237261214156329209068104310, −16.80619228902294691006193847693, −15.27451433761165154309998100791, −14.198640256400056312899002298075, −12.96451213138219517006268449095, −12.208983560931118953909044853384, −10.82394716165094679768161103585, −9.55335524912282323295813576583, −8.54050142197155784609331305883, −7.85808135720847097829769933129, −5.83995369832252540026800815709, −3.868789648886395736836210209041, −2.84825592731097388141068502234, −1.54669719908699337743019671743, 1.70032990617069532181482732322, 3.888825212390810721720376545056, 4.863289122144121464710192537603, 6.8969248661648799433229533012, 7.49960292134606028283969422870, 8.918876567880722626651802785689, 9.67859129120499980244253771669, 10.86280348254963788810770601946, 12.93747337003145703831928385795, 14.134629719278438832158581938521, 14.60768289402334270004947763097, 15.915777624847398639744964324675, 16.78510924628641054327458507918, 17.95110756570439161392318494907, 19.18619292369191165187778143224, 19.987497740257195507372891708942, 21.08234902101288502013531782401, 22.53347374958893920811895992631, 23.63265708304854200554671984976, 24.52018410703674215696434248513, 25.7608203135735083326283690359, 26.102170171597545056498568860772, 27.27461167053336546402224654739, 27.92335739415951664164956648931, 29.49924123700595537978066449112

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