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Properties

Label 1-149-149.124-r0-0-0
Degree $1$
Conductor $149$
Sign $-0.328 - 0.944i$
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  + (0.828 − 0.559i)2-s + (0.127 − 0.991i)3-s + (0.372 − 0.927i)4-s + (0.524 + 0.851i)5-s + (−0.450 − 0.892i)6-s + (−0.911 − 0.411i)7-s + (−0.210 − 0.977i)8-s + (−0.967 − 0.251i)9-s + (0.911 + 0.411i)10-s + (−0.372 − 0.927i)11-s + (−0.873 − 0.487i)12-s + (0.996 − 0.0848i)13-s + (−0.985 + 0.169i)14-s + (0.911 − 0.411i)15-s + (−0.721 − 0.691i)16-s + (−0.127 + 0.991i)17-s + ⋯
L(s)  = 1  + (0.828 − 0.559i)2-s + (0.127 − 0.991i)3-s + (0.372 − 0.927i)4-s + (0.524 + 0.851i)5-s + (−0.450 − 0.892i)6-s + (−0.911 − 0.411i)7-s + (−0.210 − 0.977i)8-s + (−0.967 − 0.251i)9-s + (0.911 + 0.411i)10-s + (−0.372 − 0.927i)11-s + (−0.873 − 0.487i)12-s + (0.996 − 0.0848i)13-s + (−0.985 + 0.169i)14-s + (0.911 − 0.411i)15-s + (−0.721 − 0.691i)16-s + (−0.127 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.328 - 0.944i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.328 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001565274 - 1.408475444i\)
\(L(\frac12)\) \(\approx\) \(1.001565274 - 1.408475444i\)
\(L(1)\) \(\approx\) \(1.276002438 - 0.9627646830i\)
\(L(1)\) \(\approx\) \(1.276002438 - 0.9627646830i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.828 - 0.559i)T \)
3 \( 1 + (0.127 - 0.991i)T \)
5 \( 1 + (0.524 + 0.851i)T \)
7 \( 1 + (-0.911 - 0.411i)T \)
11 \( 1 + (-0.372 - 0.927i)T \)
13 \( 1 + (0.996 - 0.0848i)T \)
17 \( 1 + (-0.127 + 0.991i)T \)
19 \( 1 + (0.942 + 0.333i)T \)
23 \( 1 + (0.996 + 0.0848i)T \)
29 \( 1 + (-0.594 - 0.803i)T \)
31 \( 1 + (-0.996 - 0.0848i)T \)
37 \( 1 + (0.372 + 0.927i)T \)
41 \( 1 + (0.721 + 0.691i)T \)
43 \( 1 + (0.292 + 0.956i)T \)
47 \( 1 + (0.942 - 0.333i)T \)
53 \( 1 + (-0.292 + 0.956i)T \)
59 \( 1 + (-0.778 - 0.628i)T \)
61 \( 1 + (-0.828 + 0.559i)T \)
67 \( 1 + (0.985 + 0.169i)T \)
71 \( 1 + (-0.524 - 0.851i)T \)
73 \( 1 + (0.660 - 0.750i)T \)
79 \( 1 + (-0.660 - 0.750i)T \)
83 \( 1 + (-0.660 + 0.750i)T \)
89 \( 1 + (-0.0424 - 0.999i)T \)
97 \( 1 + (0.292 - 0.956i)T \)
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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

−28.58980805928762873571137943023, −27.38247561906614975809497477227, −25.979624454106303806207882946239, −25.58364465725478067308004377311, −24.66488376272676574203666487289, −23.265940456177138447403048385997, −22.52427063115827937676728199981, −21.61427393013803004232422112882, −20.63546483386565802714733800454, −20.148067603483424358070345742848, −18.146445007182402745402568271954, −16.9462111128597194150701936219, −16.00610778219716955366467278918, −15.61879451183356679894212922046, −14.25487427496877607830869704755, −13.247487401923962546660698482594, −12.377566207668628940648151411821, −11.01323614213401973887845153794, −9.439260770886516982245807070888, −8.873248881619902148014204416139, −7.210610500592131609031512586306, −5.71015697659717455392359234, −5.052729193782733371030208566802, −3.793832213656029657638858637500, −2.552637494149984331445277978568, 1.30591940798441856142926905092, 2.83515964860394397235212907045, 3.54889439343217354481033728408, 5.85539991929215219777353046503, 6.26586040365945368146694974848, 7.52996006489124120339665062352, 9.28865928195097632641771489997, 10.63819003424422006464013377279, 11.36296400664756420736803276793, 12.88599038758895036001071174043, 13.41047571901984798063550155009, 14.19102232648169694608942981730, 15.39557600288141801632019833370, 16.79281487036150575934188440073, 18.36366714335982425604761392627, 18.87487797803623055602629534898, 19.81382721034556263622040186693, 20.94577593094264946432730436404, 22.06905924532774304082769765294, 22.94899060376263051042015826504, 23.62757750901781109507112035649, 24.76296256706848463786536631467, 25.70571185157935500760633526569, 26.60495708197631983384456199168, 28.41509944055710541768594240994

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