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Properties

Label 1-152-152.107-r0-0-0
Degree $1$
Conductor $152$
Sign $-0.0977 - 0.995i$
Analytic cond. $0.705885$
Root an. cond. $0.705885$
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  + (0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s − 7-s + (−0.5 − 0.866i)9-s + 11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 − 0.866i)29-s + 31-s + (0.5 − 0.866i)33-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s − 7-s + (−0.5 − 0.866i)9-s + 11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s − 27-s + (−0.5 − 0.866i)29-s + 31-s + (0.5 − 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.0977 - 0.995i$
Analytic conductor: \(0.705885\)
Root analytic conductor: \(0.705885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 152,\ (0:\ ),\ -0.0977 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8257953144 - 0.9108661640i\)
\(L(\frac12)\) \(\approx\) \(0.8257953144 - 0.9108661640i\)
\(L(1)\) \(\approx\) \(1.037890799 - 0.5531722440i\)
\(L(1)\) \(\approx\) \(1.037890799 - 0.5531722440i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−28.31434410357549105647708965280, −26.890206944033573488142121358115, −26.544786471549733585429670152830, −25.464180592597059360483414001276, −24.79634870152335305262641967242, −23.00023276838663618871987257960, −22.19478800047481299443907176951, −21.68772346450555481787902989593, −20.35762191198971393820385416683, −19.43698558965462839403091608217, −18.57001026705757354528684544829, −17.05568877817173706166414399073, −16.285661095789286280355040669866, −15.06946106851121444413570225761, −14.27635989363582612330175321585, −13.3926776014099634401444654854, −11.772321501525744549986596394177, −10.61632349284876406755844270679, −9.58682520765520450910480777443, −9.01542667845017877890508971854, −7.14812835749453610040334029642, −6.24056874369908183306854486676, −4.61513805801410232939481757996, −3.36484014326040880869171846597, −2.357642066886381512117172695111, 1.06790722963546637743626535438, 2.51881649374690935933780053936, 3.936167368928725345604960953830, 5.736126303687456700849377301554, 6.62900419327903790987741168732, 7.97784362168114342715596410111, 9.08263412975814841636890051849, 9.86035875503758766117611610379, 11.7162486212093102694749646002, 12.84031014063003643301509961058, 13.21784727048647127852669593264, 14.49060601031081352964076363451, 15.667993637981657776875857236706, 17.10054543238903522316281414843, 17.58071935703127549766428130732, 19.17441800183674730967353386885, 19.69059158638698073741022898677, 20.615289045881211850662105630252, 21.916582061506736551188636802431, 22.94868727961195349525175931236, 24.12256154247257722431751255026, 24.96368428950826514145678730591, 25.48430618554782836023482770258, 26.61520505364004980188404024730, 27.97609404333288980005199467299

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