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Properties

Label 1-124-124.107-r1-0-0
Degree $1$
Conductor $124$
Sign $-0.570 + 0.821i$
Analytic cond. $13.3256$
Root an. cond. $13.3256$
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  + (0.104 + 0.994i)3-s + (−0.5 − 0.866i)5-s + (0.978 + 0.207i)7-s + (−0.978 + 0.207i)9-s + (−0.669 + 0.743i)11-s + (0.913 + 0.406i)13-s + (0.809 − 0.587i)15-s + (0.669 + 0.743i)17-s + (−0.913 + 0.406i)19-s + (−0.104 + 0.994i)21-s + (−0.309 + 0.951i)23-s + (−0.5 + 0.866i)25-s + (−0.309 − 0.951i)27-s + (−0.809 − 0.587i)29-s + (−0.809 − 0.587i)33-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)3-s + (−0.5 − 0.866i)5-s + (0.978 + 0.207i)7-s + (−0.978 + 0.207i)9-s + (−0.669 + 0.743i)11-s + (0.913 + 0.406i)13-s + (0.809 − 0.587i)15-s + (0.669 + 0.743i)17-s + (−0.913 + 0.406i)19-s + (−0.104 + 0.994i)21-s + (−0.309 + 0.951i)23-s + (−0.5 + 0.866i)25-s + (−0.309 − 0.951i)27-s + (−0.809 − 0.587i)29-s + (−0.809 − 0.587i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(124\)    =    \(2^{2} \cdot 31\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(13.3256\)
Root analytic conductor: \(13.3256\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{124} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 124,\ (1:\ ),\ -0.570 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6104010297 + 1.167943328i\)
\(L(\frac12)\) \(\approx\) \(0.6104010297 + 1.167943328i\)
\(L(1)\) \(\approx\) \(0.9169657584 + 0.4252706955i\)
\(L(1)\) \(\approx\) \(0.9169657584 + 0.4252706955i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.104 + 0.994i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.978 + 0.207i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.669 + 0.743i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.913 + 0.406i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−28.35277084939429033208794992456, −27.29090492563209479897318164474, −26.25659123526528541077976491892, −25.34556678278594516246838630982, −24.076533139569370732258192752510, −23.51618992717337443604029957838, −22.53508204337087231437183411711, −21.08552912323197411434877074280, −20.11635168825046668676740282698, −18.700783172780995856149822420615, −18.46320170572926088639489907404, −17.26233785484280005818292080428, −15.80538168440963511006539236714, −14.53343525918811159300359434088, −13.83473072957877963199079098702, −12.57224276777410639168286478797, −11.28245056580573997046284331820, −10.71574251608929388406550986286, −8.552205597592647771206632302793, −7.81607714540388762430351973293, −6.74165734526002712328501705945, −5.44382094995878128183915946709, −3.55674718102853926666521430429, −2.25675924094266123163567107655, −0.53049461089081908012866234559, 1.7327332654484952372538785795, 3.748891211143886403796232681891, 4.687508419318709740110084127779, 5.71623045380840880603425815408, 7.910493963107565260722106863253, 8.57648360306701480906812280685, 9.85495978886230509362982982201, 11.057949790677211851215511427064, 12.0190639543647269407692204338, 13.38338343025951179504553665154, 14.82011670942790114587710901708, 15.48351255240715103165010820707, 16.58699275797961721336686120492, 17.47074387779589673486100071910, 18.90193858723990193236789415931, 20.24387138236918787368772114939, 20.92259472850418543759418817517, 21.57800195213155505390986522624, 23.213087207290831182124072046142, 23.74655905973341718391416417773, 25.1813310904740649440241094965, 26.06349361810255789099443136549, 27.27449266471419195614927249757, 28.05143600134678334607257382792, 28.45321845350186434347913817283

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