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Properties

Label 1-149-149.96-r0-0-0
Degree $1$
Conductor $149$
Sign $0.984 - 0.175i$
Analytic cond. $0.691953$
Root an. cond. $0.691953$
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  + (−0.721 − 0.691i)2-s + (0.873 + 0.487i)3-s + (0.0424 + 0.999i)4-s + (−0.594 − 0.803i)5-s + (−0.292 − 0.956i)6-s + (−0.127 + 0.991i)7-s + (0.660 − 0.750i)8-s + (0.524 + 0.851i)9-s + (−0.127 + 0.991i)10-s + (0.0424 − 0.999i)11-s + (−0.450 + 0.892i)12-s + (0.942 − 0.333i)13-s + (0.778 − 0.628i)14-s + (−0.127 − 0.991i)15-s + (−0.996 + 0.0848i)16-s + (0.873 + 0.487i)17-s + ⋯
L(s)  = 1  + (−0.721 − 0.691i)2-s + (0.873 + 0.487i)3-s + (0.0424 + 0.999i)4-s + (−0.594 − 0.803i)5-s + (−0.292 − 0.956i)6-s + (−0.127 + 0.991i)7-s + (0.660 − 0.750i)8-s + (0.524 + 0.851i)9-s + (−0.127 + 0.991i)10-s + (0.0424 − 0.999i)11-s + (−0.450 + 0.892i)12-s + (0.942 − 0.333i)13-s + (0.778 − 0.628i)14-s + (−0.127 − 0.991i)15-s + (−0.996 + 0.0848i)16-s + (0.873 + 0.487i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.984 - 0.175i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (96, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.984 - 0.175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9750488823 - 0.08617411868i\)
\(L(\frac12)\) \(\approx\) \(0.9750488823 - 0.08617411868i\)
\(L(1)\) \(\approx\) \(0.9388177810 - 0.1090406076i\)
\(L(1)\) \(\approx\) \(0.9388177810 - 0.1090406076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.721 - 0.691i)T \)
3 \( 1 + (0.873 + 0.487i)T \)
5 \( 1 + (-0.594 - 0.803i)T \)
7 \( 1 + (-0.127 + 0.991i)T \)
11 \( 1 + (0.0424 - 0.999i)T \)
13 \( 1 + (0.942 - 0.333i)T \)
17 \( 1 + (0.873 + 0.487i)T \)
19 \( 1 + (0.210 + 0.977i)T \)
23 \( 1 + (0.942 + 0.333i)T \)
29 \( 1 + (-0.828 - 0.559i)T \)
31 \( 1 + (0.942 + 0.333i)T \)
37 \( 1 + (0.0424 - 0.999i)T \)
41 \( 1 + (-0.996 + 0.0848i)T \)
43 \( 1 + (0.372 - 0.927i)T \)
47 \( 1 + (0.210 - 0.977i)T \)
53 \( 1 + (0.372 + 0.927i)T \)
59 \( 1 + (-0.911 + 0.411i)T \)
61 \( 1 + (-0.721 - 0.691i)T \)
67 \( 1 + (0.778 + 0.628i)T \)
71 \( 1 + (-0.594 - 0.803i)T \)
73 \( 1 + (-0.967 + 0.251i)T \)
79 \( 1 + (-0.967 - 0.251i)T \)
83 \( 1 + (-0.967 + 0.251i)T \)
89 \( 1 + (0.985 - 0.169i)T \)
97 \( 1 + (0.372 + 0.927i)T \)
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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

−27.78849417365607554681877629172, −26.89937034828568110631655855091, −25.953609764763939037720446935698, −25.69726303191503744810725232386, −24.30431966792969521180036174119, −23.39831070133648631245686235169, −22.802962512109231504050829309828, −20.70697303277834897313244495149, −19.986867686660040803673496178794, −19.01213158349159421269891488849, −18.344386802178541786379908738238, −17.260225692202977358156923900820, −15.93628569321186629261774179239, −15.01592457880336054627687304842, −14.21013123756151039336320313401, −13.237969284645015489543865046754, −11.501532311847481769171896510955, −10.31511026312173397224913901233, −9.30640024731013809434600053575, −8.01079101328272416353669864012, −7.187897352763025242745588736579, −6.59546151880249628238614440763, −4.45186403429158715004433370644, −3.0018128262550843885650835029, −1.25598888712818352197514159361, 1.45582354007115854435254379258, 3.10140121065422110599646917389, 3.87236132413931778063812671220, 5.56534508890661573747710424193, 7.76748696016491873827157498453, 8.54422565904662345349450876794, 9.127235897314604227180193971734, 10.40865831962340647725483753980, 11.599211070757263593360557673800, 12.6350521850045537470061427256, 13.638026020023144246851287587594, 15.264148925087762210913649388228, 16.08292875771912875359703286051, 16.91395637457852129517149962308, 18.754912319247656792669889843666, 19.00198693229908294161006692299, 20.1923113658580955531064936313, 21.04895485001051033710458028047, 21.5919448909825940843085509636, 23.0092573635977623311999638545, 24.70105401665775539430921442681, 25.21859102675373889631163652762, 26.34918409133344688441774038222, 27.3183394747740516936148048769, 27.91451299026391328831011428640

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