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Properties

Label 1-151-151.40-r0-0-0
Degree $1$
Conductor $151$
Sign $0.986 - 0.161i$
Analytic cond. $0.701241$
Root an. cond. $0.701241$
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  + (−0.104 + 0.994i)2-s + (0.728 − 0.684i)3-s + (−0.978 − 0.207i)4-s + (−0.895 + 0.444i)5-s + (0.604 + 0.796i)6-s + (0.463 − 0.886i)7-s + (0.309 − 0.951i)8-s + (0.0627 − 0.998i)9-s + (−0.348 − 0.937i)10-s + (0.228 − 0.973i)11-s + (−0.855 + 0.518i)12-s + (0.996 + 0.0836i)13-s + (0.832 + 0.553i)14-s + (−0.348 + 0.937i)15-s + (0.913 + 0.406i)16-s + (−0.999 − 0.0418i)17-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (0.728 − 0.684i)3-s + (−0.978 − 0.207i)4-s + (−0.895 + 0.444i)5-s + (0.604 + 0.796i)6-s + (0.463 − 0.886i)7-s + (0.309 − 0.951i)8-s + (0.0627 − 0.998i)9-s + (−0.348 − 0.937i)10-s + (0.228 − 0.973i)11-s + (−0.855 + 0.518i)12-s + (0.996 + 0.0836i)13-s + (0.832 + 0.553i)14-s + (−0.348 + 0.937i)15-s + (0.913 + 0.406i)16-s + (−0.999 − 0.0418i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(151\)
Sign: $0.986 - 0.161i$
Analytic conductor: \(0.701241\)
Root analytic conductor: \(0.701241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{151} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 151,\ (0:\ ),\ 0.986 - 0.161i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076020413 - 0.08764740232i\)
\(L(\frac12)\) \(\approx\) \(1.076020413 - 0.08764740232i\)
\(L(1)\) \(\approx\) \(1.047795944 + 0.08641493541i\)
\(L(1)\) \(\approx\) \(1.047795944 + 0.08641493541i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad151 \( 1 \)
good2 \( 1 + (-0.104 + 0.994i)T \)
3 \( 1 + (0.728 - 0.684i)T \)
5 \( 1 + (-0.895 + 0.444i)T \)
7 \( 1 + (0.463 - 0.886i)T \)
11 \( 1 + (0.228 - 0.973i)T \)
13 \( 1 + (0.996 + 0.0836i)T \)
17 \( 1 + (-0.999 - 0.0418i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.669 - 0.743i)T \)
29 \( 1 + (-0.187 - 0.982i)T \)
31 \( 1 + (0.783 + 0.621i)T \)
37 \( 1 + (-0.570 + 0.821i)T \)
41 \( 1 + (-0.992 + 0.125i)T \)
43 \( 1 + (0.463 + 0.886i)T \)
47 \( 1 + (0.387 - 0.921i)T \)
53 \( 1 + (0.876 - 0.481i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.957 + 0.289i)T \)
67 \( 1 + (0.0627 + 0.998i)T \)
71 \( 1 + (-0.999 + 0.0418i)T \)
73 \( 1 + (0.535 + 0.844i)T \)
79 \( 1 + (-0.929 - 0.368i)T \)
83 \( 1 + (0.968 + 0.248i)T \)
89 \( 1 + (-0.699 + 0.714i)T \)
97 \( 1 + (0.832 - 0.553i)T \)
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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

−27.96258235666997530331806655813, −27.47159500194630893099180792898, −26.3828839439463002074443964458, −25.39993019823852552895538783028, −24.12661732566787910354934248041, −22.8558460921859526309680687982, −21.95180002979518311888419761944, −20.93601659934148202775739588646, −20.23893389494843085826847271024, −19.48089685802346224962772305066, −18.43417281791210512915288369974, −17.26853079355012109236918157741, −15.665318009845737853701836276734, −15.14449707550640897761007360649, −13.74809301160532658628983155581, −12.67830557936420738463067339960, −11.55025674474825140245752353799, −10.747158653525499839347216773318, −9.113133839907699314473659152228, −8.85147881519323629089894623988, −7.58901135488296739744091691729, −5.09855611702894503768118316589, −4.29252390843545906101341590673, −3.12779765332487412500959036612, −1.78628778547223051672218463848, 1.0269858031659595271600161300, 3.37429712349500211932845876285, 4.30286891284137361206083373849, 6.26936584202907063463956143467, 7.08627262859952051485051645558, 8.166780106900742226648976666394, 8.67831764713220934522786158761, 10.429324641415644198993049133150, 11.70936701259146404713489961131, 13.30779402411603422658068620595, 13.93700062070051336948856022893, 14.87065637183050389173048644181, 15.85688452993885556016898968296, 16.967968156157034016232335514275, 18.222683784446688217281080246032, 18.886893872126308807634009422624, 19.8343915267644591519038574676, 21.00377934609678417502015871622, 22.67590250567017485056698222740, 23.38956925866525271079522357703, 24.23800454218304288370897691904, 24.92051254696836504915226983900, 26.35782950570058444313126400285, 26.61770763278945747803845462419, 27.54272625191981209461545307550

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