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Properties

Label 1-137-137.50-r0-0-0
Degree $1$
Conductor $137$
Sign $0.277 + 0.960i$
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  + (0.0922 − 0.995i)2-s + (0.445 + 0.895i)3-s + (−0.982 − 0.183i)4-s + (0.0922 + 0.995i)5-s + (0.932 − 0.361i)6-s + (−0.850 + 0.526i)7-s + (−0.273 + 0.961i)8-s + (−0.602 + 0.798i)9-s + 10-s + (−0.982 − 0.183i)11-s + (−0.273 − 0.961i)12-s + (−0.850 + 0.526i)13-s + (0.445 + 0.895i)14-s + (−0.850 + 0.526i)15-s + (0.932 + 0.361i)16-s + (−0.273 − 0.961i)17-s + ⋯
L(s)  = 1  + (0.0922 − 0.995i)2-s + (0.445 + 0.895i)3-s + (−0.982 − 0.183i)4-s + (0.0922 + 0.995i)5-s + (0.932 − 0.361i)6-s + (−0.850 + 0.526i)7-s + (−0.273 + 0.961i)8-s + (−0.602 + 0.798i)9-s + 10-s + (−0.982 − 0.183i)11-s + (−0.273 − 0.961i)12-s + (−0.850 + 0.526i)13-s + (0.445 + 0.895i)14-s + (−0.850 + 0.526i)15-s + (0.932 + 0.361i)16-s + (−0.273 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.277 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6762588647 + 0.5085038674i\)
\(L(\frac12)\) \(\approx\) \(0.6762588647 + 0.5085038674i\)
\(L(1)\) \(\approx\) \(0.9057086935 + 0.1527021358i\)
\(L(1)\) \(\approx\) \(0.9057086935 + 0.1527021358i\)

Euler product

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

−28.47181035570326891735832880457, −26.868922186474807934793041855184, −26.0948127017439120346173772470, −25.15767474719057115068668167080, −24.40529219443099210828988679450, −23.58888409168749778897486975518, −22.78637585940154344580241414689, −21.31187921268904428020373643888, −19.98814065308717661752422189888, −19.24774157029555876052613059042, −17.85527894407425627319686482656, −17.18620138590755743822369637728, −16.047236269022855956312177548331, −15.04845661085802626056529660517, −13.66676162810300525025702314990, −13.01407350724679510074317353893, −12.37379906138691755710309774116, −10.067344339387313672116422287563, −8.95628407681878557180233177216, −7.9466044148516037058575402285, −7.04902361877009742696643312996, −5.83172281556085044000972997052, −4.58042089530393771033736500394, −2.91568093397320579769609716516, −0.69387441295371880030711249385, 2.69054479296146323937154020977, 2.936816476677879856725645255, 4.52560064524363555089635602463, 5.78327299368661320272201580474, 7.6587960049787223431647593498, 9.25350277698470164278441146739, 9.85229191531452634641245815367, 10.8301991159990204017564159873, 11.88007256430726187830372005109, 13.32426886753275695517405203871, 14.236023691831159320306032094218, 15.23335729492796352279418174115, 16.32775016565399134943788036559, 17.92689350127639017171036730334, 18.93255060384737230766146872284, 19.60069364305561760302854372960, 20.86596757547192090309058107985, 21.63874877139697818597382122028, 22.44474016393120283685272803933, 23.11383212787776322091385499324, 24.97587474924460722547664110327, 26.24824544343127586332428475452, 26.66710340589967331893427418778, 27.64505373338538888328520537780, 29.060333650782728075612038121387

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