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Properties

Label 1-149-149.67-r0-0-0
Degree $1$
Conductor $149$
Sign $-0.0952 + 0.995i$
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  + (0.210 − 0.977i)2-s + (0.372 + 0.927i)3-s + (−0.911 − 0.411i)4-s + (−0.996 − 0.0848i)5-s + (0.985 − 0.169i)6-s + (−0.292 + 0.956i)7-s + (−0.594 + 0.803i)8-s + (−0.721 + 0.691i)9-s + (−0.292 + 0.956i)10-s + (−0.911 + 0.411i)11-s + (0.0424 − 0.999i)12-s + (−0.967 − 0.251i)13-s + (0.873 + 0.487i)14-s + (−0.292 − 0.956i)15-s + (0.660 + 0.750i)16-s + (0.372 + 0.927i)17-s + ⋯
L(s)  = 1  + (0.210 − 0.977i)2-s + (0.372 + 0.927i)3-s + (−0.911 − 0.411i)4-s + (−0.996 − 0.0848i)5-s + (0.985 − 0.169i)6-s + (−0.292 + 0.956i)7-s + (−0.594 + 0.803i)8-s + (−0.721 + 0.691i)9-s + (−0.292 + 0.956i)10-s + (−0.911 + 0.411i)11-s + (0.0424 − 0.999i)12-s + (−0.967 − 0.251i)13-s + (0.873 + 0.487i)14-s + (−0.292 − 0.956i)15-s + (0.660 + 0.750i)16-s + (0.372 + 0.927i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0952 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0952 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.0952 + 0.995i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.0952 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3690802507 + 0.4060736052i\)
\(L(\frac12)\) \(\approx\) \(0.3690802507 + 0.4060736052i\)
\(L(1)\) \(\approx\) \(0.7452191400 + 0.05903426039i\)
\(L(1)\) \(\approx\) \(0.7452191400 + 0.05903426039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.210 - 0.977i)T \)
3 \( 1 + (0.372 + 0.927i)T \)
5 \( 1 + (-0.996 - 0.0848i)T \)
7 \( 1 + (-0.292 + 0.956i)T \)
11 \( 1 + (-0.911 + 0.411i)T \)
13 \( 1 + (-0.967 - 0.251i)T \)
17 \( 1 + (0.372 + 0.927i)T \)
19 \( 1 + (0.524 - 0.851i)T \)
23 \( 1 + (-0.967 + 0.251i)T \)
29 \( 1 + (0.942 + 0.333i)T \)
31 \( 1 + (-0.967 + 0.251i)T \)
37 \( 1 + (-0.911 + 0.411i)T \)
41 \( 1 + (0.660 + 0.750i)T \)
43 \( 1 + (0.778 - 0.628i)T \)
47 \( 1 + (0.524 + 0.851i)T \)
53 \( 1 + (0.778 + 0.628i)T \)
59 \( 1 + (-0.450 - 0.892i)T \)
61 \( 1 + (0.210 - 0.977i)T \)
67 \( 1 + (0.873 - 0.487i)T \)
71 \( 1 + (-0.996 - 0.0848i)T \)
73 \( 1 + (-0.828 + 0.559i)T \)
79 \( 1 + (-0.828 - 0.559i)T \)
83 \( 1 + (-0.828 + 0.559i)T \)
89 \( 1 + (-0.127 + 0.991i)T \)
97 \( 1 + (0.778 + 0.628i)T \)
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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

−27.41795618090708654022769820371, −26.58873193884953072983305845496, −25.97979070327607042839449025422, −24.65887033522734999868209750982, −24.0051343284884482054619234877, −23.22202532029583016299924367207, −22.53899338055238252275195490932, −20.80737023286935895260041076975, −19.690889467794347275172372354567, −18.78734032527655528962982550426, −17.89231416403274605354767871366, −16.606875113251024639330260059881, −15.86750502562843462822949048475, −14.48089008739919311663831957987, −13.87478919668863368255468505091, −12.71420247796365239316869211273, −11.848844473048820377737310150236, −10.073555278104389425648728563394, −8.57272039901795162738048466786, −7.44264505492006506657641335033, −7.2749671230783122143304577034, −5.68539388614461084350729020864, −4.16219722061743886412887766783, −2.989477482703718024786841748597, −0.41045946444440664455017186928, 2.43330309793950348488853158932, 3.35092594263915164077764016299, 4.60493355253017442300696365836, 5.47583126963947743904048287473, 7.83383775332719929033161521323, 8.84833963043906027020084573904, 9.89163188279493099359140298225, 10.83596401680920653847839773045, 12.01317490625950938093796882568, 12.76117172952064853208693236457, 14.312364409959398879580273623717, 15.257568099457175033310185843016, 15.918893804988853868078609553048, 17.54889379957638546984960613546, 18.83745164731271348053562612015, 19.69956013093317631219555651245, 20.34826286287143996432239995660, 21.58606537453321717627648122532, 22.1126132935446948128680813826, 23.1752677394506781950740132093, 24.22852027850182069666115180893, 25.82320217504246533258060239996, 26.64491822688132818170095937974, 27.714292824102735019264973721633, 28.160033594066124873939018527112

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