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Properties

Label 1-149-149.113-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: $\chi_{149} (113, \cdot )$
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

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

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