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Properties

Label 1-149-149.120-r0-0-0
Degree $1$
Conductor $149$
Sign $-0.492 + 0.870i$
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  + (−0.372 + 0.927i)2-s + (0.967 + 0.251i)3-s + (−0.721 − 0.691i)4-s + (−0.450 + 0.892i)5-s + (−0.594 + 0.803i)6-s + (0.660 + 0.750i)7-s + (0.911 − 0.411i)8-s + (0.873 + 0.487i)9-s + (−0.660 − 0.750i)10-s + (0.721 − 0.691i)11-s + (−0.524 − 0.851i)12-s + (−0.985 + 0.169i)13-s + (−0.942 + 0.333i)14-s + (−0.660 + 0.750i)15-s + (0.0424 + 0.999i)16-s + (−0.967 − 0.251i)17-s + ⋯
L(s)  = 1  + (−0.372 + 0.927i)2-s + (0.967 + 0.251i)3-s + (−0.721 − 0.691i)4-s + (−0.450 + 0.892i)5-s + (−0.594 + 0.803i)6-s + (0.660 + 0.750i)7-s + (0.911 − 0.411i)8-s + (0.873 + 0.487i)9-s + (−0.660 − 0.750i)10-s + (0.721 − 0.691i)11-s + (−0.524 − 0.851i)12-s + (−0.985 + 0.169i)13-s + (−0.942 + 0.333i)14-s + (−0.660 + 0.750i)15-s + (0.0424 + 0.999i)16-s + (−0.967 − 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.492 + 0.870i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (120, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5747727386 + 0.9853928580i\)
\(L(\frac12)\) \(\approx\) \(0.5747727386 + 0.9853928580i\)
\(L(1)\) \(\approx\) \(0.8389323718 + 0.6977994979i\)
\(L(1)\) \(\approx\) \(0.8389323718 + 0.6977994979i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.372 + 0.927i)T \)
3 \( 1 + (0.967 + 0.251i)T \)
5 \( 1 + (-0.450 + 0.892i)T \)
7 \( 1 + (0.660 + 0.750i)T \)
11 \( 1 + (0.721 - 0.691i)T \)
13 \( 1 + (-0.985 + 0.169i)T \)
17 \( 1 + (-0.967 - 0.251i)T \)
19 \( 1 + (0.778 + 0.628i)T \)
23 \( 1 + (-0.985 - 0.169i)T \)
29 \( 1 + (-0.292 + 0.956i)T \)
31 \( 1 + (0.985 + 0.169i)T \)
37 \( 1 + (-0.721 + 0.691i)T \)
41 \( 1 + (-0.0424 - 0.999i)T \)
43 \( 1 + (0.828 - 0.559i)T \)
47 \( 1 + (0.778 - 0.628i)T \)
53 \( 1 + (-0.828 - 0.559i)T \)
59 \( 1 + (-0.210 - 0.977i)T \)
61 \( 1 + (0.372 - 0.927i)T \)
67 \( 1 + (0.942 + 0.333i)T \)
71 \( 1 + (0.450 - 0.892i)T \)
73 \( 1 + (-0.127 - 0.991i)T \)
79 \( 1 + (0.127 - 0.991i)T \)
83 \( 1 + (0.127 + 0.991i)T \)
89 \( 1 + (0.996 - 0.0848i)T \)
97 \( 1 + (0.828 + 0.559i)T \)
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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

−27.74732253002222531409195324692, −26.89016839010850745888314315610, −26.21190888022559695634605820094, −24.75074248152323928276909774148, −24.12846667044637132263833099356, −22.73866302778200834854147900728, −21.48583734467560213840803005101, −20.41087580097325135973243682645, −19.98124122183988891656121994461, −19.3337684970730878461060614752, −17.76953133190128529030255382512, −17.19649491730190162757208614567, −15.64563673533192069078402994038, −14.31408656753304931752199153411, −13.393197728235739313750502538476, −12.39142979874593607838664669392, −11.500169887479358711295870645022, −9.945093971937083418084727218210, −9.14219841751241117219234614341, −8.03094366463317268350226980709, −7.30902329489965420929838055309, −4.597710777184411624739194542784, −4.02805712479464977473305393185, −2.3440959319315332427559353430, −1.1459729689645882422776765493, 2.088895541208975933263223789886, 3.66655556714588306465876758785, 4.97963728079739429685731645665, 6.53445034092975002506006221793, 7.608196228913505947897015025066, 8.50036573555292873433615439300, 9.458904463328184721528018123495, 10.64302071615537813606814286124, 12.03842921916654603433465451327, 13.97627378912270250267762149493, 14.333512551161165863555725731786, 15.3263223530957175205856298203, 16.03126770146680039033587404341, 17.51620681063144519213746844966, 18.60326545197289468398418464318, 19.20393229024497248244018040178, 20.25820118674128713346799298716, 21.980331043176635450203117840460, 22.27788208857889651745233327404, 24.08372417273113414622505815271, 24.5677033542782652286842935083, 25.54204991597750310305001078521, 26.65812993557579228068433602869, 27.02172264099526688517563663471, 27.89535932259692776879919887005

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