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Properties

Label 1-149-149.132-r0-0-0
Degree $1$
Conductor $149$
Sign $0.176 + 0.984i$
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  + (−0.524 + 0.851i)2-s + (−0.778 + 0.628i)3-s + (−0.450 − 0.892i)4-s + (0.660 + 0.750i)5-s + (−0.127 − 0.991i)6-s + (0.985 − 0.169i)7-s + (0.996 + 0.0848i)8-s + (0.210 − 0.977i)9-s + (−0.985 + 0.169i)10-s + (0.450 − 0.892i)11-s + (0.911 + 0.411i)12-s + (0.828 − 0.559i)13-s + (−0.372 + 0.927i)14-s + (−0.985 − 0.169i)15-s + (−0.594 + 0.803i)16-s + (0.778 − 0.628i)17-s + ⋯
L(s)  = 1  + (−0.524 + 0.851i)2-s + (−0.778 + 0.628i)3-s + (−0.450 − 0.892i)4-s + (0.660 + 0.750i)5-s + (−0.127 − 0.991i)6-s + (0.985 − 0.169i)7-s + (0.996 + 0.0848i)8-s + (0.210 − 0.977i)9-s + (−0.985 + 0.169i)10-s + (0.450 − 0.892i)11-s + (0.911 + 0.411i)12-s + (0.828 − 0.559i)13-s + (−0.372 + 0.927i)14-s + (−0.985 − 0.169i)15-s + (−0.594 + 0.803i)16-s + (0.778 − 0.628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.176 + 0.984i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.176 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6361606002 + 0.5322992051i\)
\(L(\frac12)\) \(\approx\) \(0.6361606002 + 0.5322992051i\)
\(L(1)\) \(\approx\) \(0.7019891431 + 0.4101529708i\)
\(L(1)\) \(\approx\) \(0.7019891431 + 0.4101529708i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.524 + 0.851i)T \)
3 \( 1 + (-0.778 + 0.628i)T \)
5 \( 1 + (0.660 + 0.750i)T \)
7 \( 1 + (0.985 - 0.169i)T \)
11 \( 1 + (0.450 - 0.892i)T \)
13 \( 1 + (0.828 - 0.559i)T \)
17 \( 1 + (0.778 - 0.628i)T \)
19 \( 1 + (-0.721 + 0.691i)T \)
23 \( 1 + (0.828 + 0.559i)T \)
29 \( 1 + (-0.967 - 0.251i)T \)
31 \( 1 + (-0.828 - 0.559i)T \)
37 \( 1 + (-0.450 + 0.892i)T \)
41 \( 1 + (0.594 - 0.803i)T \)
43 \( 1 + (-0.873 + 0.487i)T \)
47 \( 1 + (-0.721 - 0.691i)T \)
53 \( 1 + (0.873 + 0.487i)T \)
59 \( 1 + (-0.0424 + 0.999i)T \)
61 \( 1 + (0.524 - 0.851i)T \)
67 \( 1 + (0.372 + 0.927i)T \)
71 \( 1 + (-0.660 - 0.750i)T \)
73 \( 1 + (0.942 + 0.333i)T \)
79 \( 1 + (-0.942 + 0.333i)T \)
83 \( 1 + (-0.942 - 0.333i)T \)
89 \( 1 + (0.292 + 0.956i)T \)
97 \( 1 + (-0.873 - 0.487i)T \)
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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

−28.15094228332871383913892523324, −27.507958288813026549593053264085, −25.8710064958330595536172207024, −25.049091690804792337880962221610, −23.94320625580327649178942050759, −22.951513316683705291137401978274, −21.641651769475016313248958121412, −21.07062617464574294272314376615, −19.965511573840063763740517943211, −18.7620474006621782271867582081, −17.878694692132338581875612155699, −17.20988906944645556137645581644, −16.44524011639542785248212684864, −14.453676679679862530438383855225, −13.13151664438662690911108924081, −12.489937682770936159550706923898, −11.438316779239329161277829932114, −10.56836630541283177420416183530, −9.18293399749586440500745830346, −8.23936889980860148742256187923, −6.89666730216160336407472469800, −5.330876555515790627204818570704, −4.30096265923322393113081565966, −1.991684419733439723768667155855, −1.33467405867698974522236182624, 1.31176685782911552483512238153, 3.721324296868165316816718384668, 5.3451801688726313623587192026, 5.93874012175550956723507930781, 7.16628880505679663847776710909, 8.537953958322415367284614422698, 9.73087364072330598642687891791, 10.74219949108667161645733443466, 11.376998795180431437734869582712, 13.44590568778272631589934466275, 14.52882038029331498989213537640, 15.21229467688891910381425576919, 16.56657830193491672108635620608, 17.17419091392349304409405643077, 18.17818602009325556432682593856, 18.81043635628944951243400703081, 20.64668553122737327097742940413, 21.540056600224047992052925517589, 22.67866260344927110198794695668, 23.36928200461052289646194129195, 24.501099102531285263218971437357, 25.50572592584154833583066376507, 26.49298487299478088885057863736, 27.46596035377685179195012784380, 27.77389625289796144057986613573

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