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Properties

Label 1-149-149.20-r0-0-0
Degree $1$
Conductor $149$
Sign $0.624 - 0.781i$
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  + (−0.210 − 0.977i)2-s + (−0.372 + 0.927i)3-s + (−0.911 + 0.411i)4-s + (−0.996 + 0.0848i)5-s + (0.985 + 0.169i)6-s + (−0.292 − 0.956i)7-s + (0.594 + 0.803i)8-s + (−0.721 − 0.691i)9-s + (0.292 + 0.956i)10-s + (0.911 + 0.411i)11-s + (−0.0424 − 0.999i)12-s + (0.967 − 0.251i)13-s + (−0.873 + 0.487i)14-s + (0.292 − 0.956i)15-s + (0.660 − 0.750i)16-s + (0.372 − 0.927i)17-s + ⋯
L(s)  = 1  + (−0.210 − 0.977i)2-s + (−0.372 + 0.927i)3-s + (−0.911 + 0.411i)4-s + (−0.996 + 0.0848i)5-s + (0.985 + 0.169i)6-s + (−0.292 − 0.956i)7-s + (0.594 + 0.803i)8-s + (−0.721 − 0.691i)9-s + (0.292 + 0.956i)10-s + (0.911 + 0.411i)11-s + (−0.0424 − 0.999i)12-s + (0.967 − 0.251i)13-s + (−0.873 + 0.487i)14-s + (0.292 − 0.956i)15-s + (0.660 − 0.750i)16-s + (0.372 − 0.927i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.624 - 0.781i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.624 - 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6079694724 - 0.2923434325i\)
\(L(\frac12)\) \(\approx\) \(0.6079694724 - 0.2923434325i\)
\(L(1)\) \(\approx\) \(0.6824103396 - 0.1968237654i\)
\(L(1)\) \(\approx\) \(0.6824103396 - 0.1968237654i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.210 - 0.977i)T \)
3 \( 1 + (-0.372 + 0.927i)T \)
5 \( 1 + (-0.996 + 0.0848i)T \)
7 \( 1 + (-0.292 - 0.956i)T \)
11 \( 1 + (0.911 + 0.411i)T \)
13 \( 1 + (0.967 - 0.251i)T \)
17 \( 1 + (0.372 - 0.927i)T \)
19 \( 1 + (0.524 + 0.851i)T \)
23 \( 1 + (0.967 + 0.251i)T \)
29 \( 1 + (0.942 - 0.333i)T \)
31 \( 1 + (-0.967 - 0.251i)T \)
37 \( 1 + (-0.911 - 0.411i)T \)
41 \( 1 + (-0.660 + 0.750i)T \)
43 \( 1 + (-0.778 - 0.628i)T \)
47 \( 1 + (0.524 - 0.851i)T \)
53 \( 1 + (0.778 - 0.628i)T \)
59 \( 1 + (0.450 - 0.892i)T \)
61 \( 1 + (0.210 + 0.977i)T \)
67 \( 1 + (0.873 + 0.487i)T \)
71 \( 1 + (0.996 - 0.0848i)T \)
73 \( 1 + (-0.828 - 0.559i)T \)
79 \( 1 + (0.828 - 0.559i)T \)
83 \( 1 + (0.828 + 0.559i)T \)
89 \( 1 + (0.127 + 0.991i)T \)
97 \( 1 + (-0.778 + 0.628i)T \)
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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

−28.09325077969650672489632883177, −27.4077223700593446058177625225, −26.05004138864317437742280402046, −25.177675823857943112607714799135, −24.3133600555217537166537708935, −23.58690491459342714156627285696, −22.728493671759401055285740468145, −21.81155917829063825259901374908, −19.75744095282950213036116314354, −19.020775052525935212018385942621, −18.36949536470173574607655769223, −17.15358501416028757430044839250, −16.23303267398612046842536213321, −15.31028992866964897858172528660, −14.184148731252812126122500423981, −12.982354542077544965419580691512, −12.03083189218604556612158354368, −10.932372409440473574359780212477, −8.90738379730118029923355412571, −8.41346081723964471180050924408, −7.05615863668170225483181875316, −6.277694064852569542209007139226, −5.11092254178672962818262611534, −3.46358709253603913926486747629, −1.15756384185786381647360948387, 0.89456756260427096908350443546, 3.42169371780441712481802895446, 3.84697089067070686756940900758, 5.0813653185039781159761281359, 7.0158254355405882269046996571, 8.435619585351336980563882830842, 9.58507942141317098458432002072, 10.529330829452689194066668654226, 11.427733437140948078456886164575, 12.186683104376955542575744891854, 13.67608172559074893593642214658, 14.79270333049195281806523732247, 16.1633731936862231544940490436, 16.87268008725384784587088977587, 18.071481685134272984006762399616, 19.29172885517963754573373936875, 20.301756355889220345741927510715, 20.7021724894481384335492552008, 22.12194898387700147437610113577, 23.109711584626498736305348201369, 23.205207320704518791339334013125, 25.3851801562772679068680405216, 26.61093765278284270160341879223, 27.20049322697097581389040393431, 27.800150089333701933097409141611

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