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Properties

Label 1-149-149.144-r0-0-0
Degree $1$
Conductor $149$
Sign $0.693 + 0.720i$
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  + (0.292 + 0.956i)2-s + (−0.660 − 0.750i)3-s + (−0.828 + 0.559i)4-s + (0.873 + 0.487i)5-s + (0.524 − 0.851i)6-s + (0.210 − 0.977i)7-s + (−0.778 − 0.628i)8-s + (−0.127 + 0.991i)9-s + (−0.210 + 0.977i)10-s + (0.828 + 0.559i)11-s + (0.967 + 0.251i)12-s + (−0.0424 − 0.999i)13-s + (0.996 − 0.0848i)14-s + (−0.210 − 0.977i)15-s + (0.372 − 0.927i)16-s + (0.660 + 0.750i)17-s + ⋯
L(s)  = 1  + (0.292 + 0.956i)2-s + (−0.660 − 0.750i)3-s + (−0.828 + 0.559i)4-s + (0.873 + 0.487i)5-s + (0.524 − 0.851i)6-s + (0.210 − 0.977i)7-s + (−0.778 − 0.628i)8-s + (−0.127 + 0.991i)9-s + (−0.210 + 0.977i)10-s + (0.828 + 0.559i)11-s + (0.967 + 0.251i)12-s + (−0.0424 − 0.999i)13-s + (0.996 − 0.0848i)14-s + (−0.210 − 0.977i)15-s + (0.372 − 0.927i)16-s + (0.660 + 0.750i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.693 + 0.720i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.693 + 0.720i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.050649272 + 0.4471385350i\)
\(L(\frac12)\) \(\approx\) \(1.050649272 + 0.4471385350i\)
\(L(1)\) \(\approx\) \(1.041824077 + 0.3364711124i\)
\(L(1)\) \(\approx\) \(1.041824077 + 0.3364711124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.292 + 0.956i)T \)
3 \( 1 + (-0.660 - 0.750i)T \)
5 \( 1 + (0.873 + 0.487i)T \)
7 \( 1 + (0.210 - 0.977i)T \)
11 \( 1 + (0.828 + 0.559i)T \)
13 \( 1 + (-0.0424 - 0.999i)T \)
17 \( 1 + (0.660 + 0.750i)T \)
19 \( 1 + (0.985 + 0.169i)T \)
23 \( 1 + (-0.0424 + 0.999i)T \)
29 \( 1 + (-0.450 + 0.892i)T \)
31 \( 1 + (0.0424 - 0.999i)T \)
37 \( 1 + (-0.828 - 0.559i)T \)
41 \( 1 + (-0.372 + 0.927i)T \)
43 \( 1 + (0.594 - 0.803i)T \)
47 \( 1 + (0.985 - 0.169i)T \)
53 \( 1 + (-0.594 - 0.803i)T \)
59 \( 1 + (-0.942 - 0.333i)T \)
61 \( 1 + (-0.292 - 0.956i)T \)
67 \( 1 + (-0.996 - 0.0848i)T \)
71 \( 1 + (-0.873 - 0.487i)T \)
73 \( 1 + (-0.911 + 0.411i)T \)
79 \( 1 + (0.911 + 0.411i)T \)
83 \( 1 + (0.911 - 0.411i)T \)
89 \( 1 + (0.721 + 0.691i)T \)
97 \( 1 + (0.594 + 0.803i)T \)
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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

−28.26210329488826909870356684616, −27.37780835094170246786127639356, −26.38746933080497921605339156399, −24.8374472897222574778824960760, −23.98062637772637651282596323188, −22.52749201718835021773552530042, −21.99712015695956938068756347699, −21.148242007904247413242053501953, −20.54632333477120382665476446374, −19.00506635442490902868431572502, −18.08864913018776651648558534610, −17.06174177506401712247040271480, −16.01588601045394238064860549878, −14.55050000029805898844434697810, −13.78854514847187780716111229404, −12.14831096064308736811930672162, −11.80634602770625042523160138155, −10.4986103791787196965598434839, −9.29512585144248067692197261337, −8.978272096078567001277082710489, −6.22930997544603985309579999360, −5.37527933708218758137042529458, −4.427583897898210935768842634799, −2.90489973768326498435748726312, −1.32985121910295787963965513855, 1.40107994262033708655340879467, 3.51978204510808984560733623313, 5.170091681031847199763082114412, 6.03895122088506273200808328788, 7.12335659334940051593425544941, 7.78874426363376386082409669700, 9.568156135918449734886236729546, 10.67958656314693035639117299121, 12.17019790853084810514116206149, 13.23433360653799774710775556965, 13.98078944761742741372716300175, 14.92538042364723542321782286753, 16.48858731604093959677218508965, 17.421821614858442544946911011784, 17.70165824133009429386709728579, 18.91660622887448508112736420293, 20.36746797422557028314727296757, 21.86875861623183311136474738791, 22.580904869646402079389844979635, 23.34502608031369872907834818650, 24.34948600573013712580117454242, 25.22377208251322027916252167336, 25.93448034636852456355117706867, 27.194702554257593896992863723342, 28.096787562440954542341356690441

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