Properties

Label 1-137-137.73-r0-0-0
Degree $1$
Conductor $137$
Sign $0.0841 - 0.996i$
Analytic cond. $0.636225$
Root an. cond. $0.636225$
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  + (−0.850 + 0.526i)2-s + (0.932 − 0.361i)3-s + (0.445 − 0.895i)4-s + (−0.850 − 0.526i)5-s + (−0.602 + 0.798i)6-s + (−0.982 − 0.183i)7-s + (0.0922 + 0.995i)8-s + (0.739 − 0.673i)9-s + 10-s + (0.445 − 0.895i)11-s + (0.0922 − 0.995i)12-s + (−0.982 − 0.183i)13-s + (0.932 − 0.361i)14-s + (−0.982 − 0.183i)15-s + (−0.602 − 0.798i)16-s + (0.0922 − 0.995i)17-s + ⋯
L(s)  = 1  + (−0.850 + 0.526i)2-s + (0.932 − 0.361i)3-s + (0.445 − 0.895i)4-s + (−0.850 − 0.526i)5-s + (−0.602 + 0.798i)6-s + (−0.982 − 0.183i)7-s + (0.0922 + 0.995i)8-s + (0.739 − 0.673i)9-s + 10-s + (0.445 − 0.895i)11-s + (0.0922 − 0.995i)12-s + (−0.982 − 0.183i)13-s + (0.932 − 0.361i)14-s + (−0.982 − 0.183i)15-s + (−0.602 − 0.798i)16-s + (0.0922 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.0841 - 0.996i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.0841 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4819377846 - 0.4429468011i\)
\(L(\frac12)\) \(\approx\) \(0.4819377846 - 0.4429468011i\)
\(L(1)\) \(\approx\) \(0.7014835058 - 0.1796576414i\)
\(L(1)\) \(\approx\) \(0.7014835058 - 0.1796576414i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.850 + 0.526i)T \)
3 \( 1 + (0.932 - 0.361i)T \)
5 \( 1 + (-0.850 - 0.526i)T \)
7 \( 1 + (-0.982 - 0.183i)T \)
11 \( 1 + (0.445 - 0.895i)T \)
13 \( 1 + (-0.982 - 0.183i)T \)
17 \( 1 + (0.0922 - 0.995i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (-0.602 - 0.798i)T \)
29 \( 1 + (-0.602 - 0.798i)T \)
31 \( 1 + (-0.273 - 0.961i)T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + (-0.273 + 0.961i)T \)
47 \( 1 + (0.739 - 0.673i)T \)
53 \( 1 + (-0.273 + 0.961i)T \)
59 \( 1 + (0.739 - 0.673i)T \)
61 \( 1 + (0.739 + 0.673i)T \)
67 \( 1 + (-0.982 - 0.183i)T \)
71 \( 1 + (0.445 + 0.895i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (0.932 + 0.361i)T \)
83 \( 1 + (0.0922 + 0.995i)T \)
89 \( 1 + (-0.850 - 0.526i)T \)
97 \( 1 + (0.445 - 0.895i)T \)
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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

−28.46824161447259137625290646984, −27.611728031807433288202978227361, −26.74292524033977402468390469713, −25.90192002432854250924928604982, −25.36997151535717877729012553983, −23.89513932933989010960018553021, −22.25231009326558305913002006889, −21.764074534043300521795840992891, −20.21357684008385650951803517942, −19.55345231812093164848410593004, −19.16404401919506608520290644718, −17.79077760977818742851038979916, −16.450845951649497268253894806515, −15.50221698534004301542340676857, −14.675834803752481695868756755854, −12.966860975615084299451448925618, −12.10896552581777744306462875282, −10.69941737314031504892988004639, −9.74539734673350048299274164405, −8.91236123352019081022629587243, −7.601496716012889532833862272976, −6.86877926246088762271664594390, −4.22220787087649122698635594603, −3.268466861748026180867615628542, −2.14617473290641912700623200148, 0.67877200282695088954317582354, 2.61528364045656560178689990254, 4.09220328429964461973704222377, 6.02141959828550911421946063548, 7.2893752893769688134365286880, 8.05476227671273994777202117854, 9.14106877401413841128455375785, 9.93942495124779059067036835495, 11.617460727067708016754161103033, 12.78917753463144286866394690934, 14.1075896916162388185581555325, 15.084421342455098397957192329311, 16.184898752052859047859268435336, 16.79715423415620039424280419620, 18.49380991571818277428155868060, 19.1920181066680397108648110565, 19.886673674911879349971497322331, 20.68065557670679922374661201088, 22.52677421366561324726554509763, 23.694138660634789549742559114866, 24.6373620521488708137989441525, 25.14099331358703683675500884817, 26.55418484464593989169822836997, 26.85732774472093363870643855555, 28.00221163485444081324873507970

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