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Properties

Label 1-149-149.130-r0-0-0
Degree $1$
Conductor $149$
Sign $0.990 + 0.135i$
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  + (0.911 + 0.411i)2-s + (0.721 − 0.691i)3-s + (0.660 + 0.750i)4-s + (0.985 + 0.169i)5-s + (0.942 − 0.333i)6-s + (−0.828 − 0.559i)7-s + (0.292 + 0.956i)8-s + (0.0424 − 0.999i)9-s + (0.828 + 0.559i)10-s + (−0.660 + 0.750i)11-s + (0.996 + 0.0848i)12-s + (−0.873 − 0.487i)13-s + (−0.524 − 0.851i)14-s + (0.828 − 0.559i)15-s + (−0.127 + 0.991i)16-s + (−0.721 + 0.691i)17-s + ⋯
L(s)  = 1  + (0.911 + 0.411i)2-s + (0.721 − 0.691i)3-s + (0.660 + 0.750i)4-s + (0.985 + 0.169i)5-s + (0.942 − 0.333i)6-s + (−0.828 − 0.559i)7-s + (0.292 + 0.956i)8-s + (0.0424 − 0.999i)9-s + (0.828 + 0.559i)10-s + (−0.660 + 0.750i)11-s + (0.996 + 0.0848i)12-s + (−0.873 − 0.487i)13-s + (−0.524 − 0.851i)14-s + (0.828 − 0.559i)15-s + (−0.127 + 0.991i)16-s + (−0.721 + 0.691i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.990 + 0.135i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.990 + 0.135i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.264652808 + 0.1542603164i\)
\(L(\frac12)\) \(\approx\) \(2.264652808 + 0.1542603164i\)
\(L(1)\) \(\approx\) \(2.025316879 + 0.1284008209i\)
\(L(1)\) \(\approx\) \(2.025316879 + 0.1284008209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.911 + 0.411i)T \)
3 \( 1 + (0.721 - 0.691i)T \)
5 \( 1 + (0.985 + 0.169i)T \)
7 \( 1 + (-0.828 - 0.559i)T \)
11 \( 1 + (-0.660 + 0.750i)T \)
13 \( 1 + (-0.873 - 0.487i)T \)
17 \( 1 + (-0.721 + 0.691i)T \)
19 \( 1 + (-0.450 - 0.892i)T \)
23 \( 1 + (-0.873 + 0.487i)T \)
29 \( 1 + (0.778 + 0.628i)T \)
31 \( 1 + (0.873 - 0.487i)T \)
37 \( 1 + (0.660 - 0.750i)T \)
41 \( 1 + (0.127 - 0.991i)T \)
43 \( 1 + (-0.210 + 0.977i)T \)
47 \( 1 + (-0.450 + 0.892i)T \)
53 \( 1 + (0.210 + 0.977i)T \)
59 \( 1 + (0.594 - 0.803i)T \)
61 \( 1 + (-0.911 - 0.411i)T \)
67 \( 1 + (0.524 - 0.851i)T \)
71 \( 1 + (-0.985 - 0.169i)T \)
73 \( 1 + (0.372 - 0.927i)T \)
79 \( 1 + (-0.372 - 0.927i)T \)
83 \( 1 + (-0.372 + 0.927i)T \)
89 \( 1 + (0.967 + 0.251i)T \)
97 \( 1 + (-0.210 - 0.977i)T \)
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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

−28.5787951882146992298418119208, −27.048193235505632538667762651365, −25.98259271458927466757347859373, −25.01872976124451755659885348653, −24.41208758185655836343579249294, −22.81701072403508782145520394957, −21.83191327733503014997397652421, −21.44494530687593042193980650633, −20.4313718357543988861236574537, −19.43367272214790643696138108053, −18.49630166996166136687906010988, −16.58946786431750208208703985952, −15.86960523838318525251560283495, −14.73262778700382365152348766029, −13.78969145186041063952989605184, −13.11269760927835370209182354495, −11.8454702612790771999933884310, −10.26126247028166494318476626455, −9.80305545973769243853606988998, −8.515576539555836980327512185988, −6.588872096033711417472539400858, −5.506547260925636113672058494464, −4.420212711161923771250907940252, −2.89962096712324447184948767081, −2.24328239148803277801957742527, 2.13255870226544302756796645641, 2.96971996005788598350493042610, 4.51878394400087145343205440015, 6.0738523379330787752829403828, 6.89381065472436309910018414542, 7.86717709992051711763927531358, 9.399116462806855238263664785895, 10.566997738590035046776078932300, 12.434317553976351200597124062798, 13.044966018553099959796942735830, 13.77071434194515197933443673996, 14.79396914690748056025684820438, 15.72499044290324520517497246617, 17.28589068745075791032707046057, 17.834433949547935781805576157610, 19.546633011556811847949317266775, 20.21355905975226008746847986341, 21.35326666637134815089389577841, 22.2929064414853266168171491275, 23.354763681521062527221395902444, 24.2587386910953102714894372299, 25.202820334148872401557137635856, 25.99295572714000887312893466657, 26.40370036903997486552845210298, 28.62815420107254986451788968618

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