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Properties

Label 1-149-149.114-r0-0-0
Degree $1$
Conductor $149$
Sign $0.0392 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0392 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0392 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.0392 + 0.999i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.0392 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.362749677 + 1.310254914i\)
\(L(\frac12)\) \(\approx\) \(1.362749677 + 1.310254914i\)
\(L(1)\) \(\approx\) \(1.440399802 + 0.9251088592i\)
\(L(1)\) \(\approx\) \(1.440399802 + 0.9251088592i\)

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

−28.01628587057552549254125988369, −26.91202904239066931294962278042, −25.92489459814048615368657671569, −24.81169379794003340910760377199, −23.88757539053309600487729246734, −22.94841199164628319061795833131, −21.74690198891681338677574312850, −20.822561647714625403955103618, −20.22229837518521537358491287784, −18.78670597942707651166703923946, −18.389901068567531646099908860927, −17.28109701952858561186970462814, −14.9508987245712176913067588425, −14.54540544337440883173274155588, −13.722836432078060282446513409839, −12.594780247489140045179171945971, −11.62482191855566707288510777619, −10.25954511773306258794695955882, −9.49195825638520652945400831766, −7.92493584637021039430526350083, −6.75767307310485515385601117, −5.28582907725725865066229413791, −3.887494486117323360460315557629, −2.36996694051358080614289584684, −1.820713459150492373171460448425, 2.23498418198261615292332676483, 3.750059711025585479478774279271, 5.04535577055983117931992047358, 5.62805627732993147045745753503, 7.64153703496236555199467625328, 8.411652817713915844954781361872, 9.30030494032244209181365349760, 10.71745461134995312808677676394, 12.40606562698579606000601620724, 13.42064668672243222824459149541, 14.31206823062413950958874012105, 15.08647141161162259230267379702, 16.20272439034345951671550448914, 17.06527770712901054180161795322, 18.05358097960290514080116902915, 19.623408717073975710039712481364, 20.95941419665529564938031966667, 21.36709225615381343223156946021, 22.19066974972421906245602455061, 23.980047494568404213231935393585, 24.336224864097644462784304839592, 25.4375467235823325462858008010, 26.102694521310480192777792196837, 27.31036136054149583443068942413, 27.88436584438919594722802400512

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