Properties

Label 1-115-115.112-r0-0-0
Degree $1$
Conductor $115$
Sign $0.411 - 0.911i$
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  + (−0.281 − 0.959i)2-s + (−0.755 + 0.654i)3-s + (−0.841 + 0.540i)4-s + (0.841 + 0.540i)6-s + (−0.909 − 0.415i)7-s + (0.755 + 0.654i)8-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)11-s + (0.281 − 0.959i)12-s + (0.909 − 0.415i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (0.540 − 0.841i)17-s + (−0.989 + 0.142i)18-s + (0.841 − 0.540i)19-s + ⋯
L(s)  = 1  + (−0.281 − 0.959i)2-s + (−0.755 + 0.654i)3-s + (−0.841 + 0.540i)4-s + (0.841 + 0.540i)6-s + (−0.909 − 0.415i)7-s + (0.755 + 0.654i)8-s + (0.142 − 0.989i)9-s + (0.959 + 0.281i)11-s + (0.281 − 0.959i)12-s + (0.909 − 0.415i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (0.540 − 0.841i)17-s + (−0.989 + 0.142i)18-s + (0.841 − 0.540i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ 0.411 - 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5323706285 - 0.3436096007i\)
\(L(\frac12)\) \(\approx\) \(0.5323706285 - 0.3436096007i\)
\(L(1)\) \(\approx\) \(0.6460083460 - 0.2392673332i\)
\(L(1)\) \(\approx\) \(0.6460083460 - 0.2392673332i\)

Euler product

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

−29.24244376976266581174576970971, −28.30108332010842005829290045949, −27.59516001453058181444862560123, −26.183340307600938198433923722337, −25.294098322798898940009332036132, −24.45492232806362733831261537631, −23.448683121990604189983787670306, −22.61352416948139776534428813035, −21.79584437076539868975597645470, −19.69057237064235244626014246386, −18.78492558959707886387089314735, −18.08732090809913054894033018548, −16.60771872105992809789224826910, −16.41988403470798877711864642838, −14.87472609557958061305293091019, −13.61948411670766248906934016285, −12.67892757041488537720291479973, −11.364192504779434746015255957797, −9.900305759298969661731974344077, −8.72783549314365145569507502089, −7.399472008128270643695449594384, −6.24876654894646372758984495474, −5.70479529666232753333742118785, −3.86347645729603121328694079926, −1.314745966289944636978312823948, 0.925643852985296184549135749172, 3.22553722307239668937040501599, 4.145895252652475294042820317453, 5.63292532228803086748475309112, 7.16866458457861117229514760283, 9.10152475091617229291281522063, 9.78161757937078461497544708973, 10.89300828157441475517475958708, 11.815822286867463879981571330742, 12.88016890779948869382274408843, 14.116861999503751994528659976872, 15.80356193976797582307189205852, 16.73212552571081502279641439808, 17.69691022515107621219031991543, 18.75925331892166261152762451861, 20.06431310915466022545223815345, 20.731547948425013984772230603037, 22.096314199442213125168602188789, 22.60338340383062096836505982752, 23.48198493989084513952784524759, 25.381003124734005638768572908925, 26.41117266668051941007129637603, 27.35590792736875008289951768528, 28.12527214542707831683285217463, 29.03714613672458195147746489509

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