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Properties

Label 1-137-137.19-r0-0-0
Degree $1$
Conductor $137$
Sign $-0.726 - 0.686i$
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  + (−0.739 + 0.673i)2-s + (−0.526 + 0.850i)3-s + (0.0922 − 0.995i)4-s + (−0.673 + 0.739i)5-s + (−0.183 − 0.982i)6-s + (0.273 + 0.961i)7-s + (0.602 + 0.798i)8-s + (−0.445 − 0.895i)9-s − i·10-s + (−0.0922 + 0.995i)11-s + (0.798 + 0.602i)12-s + (−0.961 + 0.273i)13-s + (−0.850 − 0.526i)14-s + (−0.273 − 0.961i)15-s + (−0.982 − 0.183i)16-s + (0.602 − 0.798i)17-s + ⋯
L(s)  = 1  + (−0.739 + 0.673i)2-s + (−0.526 + 0.850i)3-s + (0.0922 − 0.995i)4-s + (−0.673 + 0.739i)5-s + (−0.183 − 0.982i)6-s + (0.273 + 0.961i)7-s + (0.602 + 0.798i)8-s + (−0.445 − 0.895i)9-s − i·10-s + (−0.0922 + 0.995i)11-s + (0.798 + 0.602i)12-s + (−0.961 + 0.273i)13-s + (−0.850 − 0.526i)14-s + (−0.273 − 0.961i)15-s + (−0.982 − 0.183i)16-s + (0.602 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.726 - 0.686i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ -0.726 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1148517740 + 0.2886614141i\)
\(L(\frac12)\) \(\approx\) \(-0.1148517740 + 0.2886614141i\)
\(L(1)\) \(\approx\) \(0.3006125245 + 0.3601143163i\)
\(L(1)\) \(\approx\) \(0.3006125245 + 0.3601143163i\)

Euler product

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

−27.80140298522476856364930202833, −27.26670982835830622306133602296, −26.11360653569303748567082783444, −24.75074413739616338993835639696, −23.97469059263140143349230789689, −23.024860884942711524384396087710, −21.727520303782854867295948860518, −20.568918087563915396114762557762, −19.536667107891584144151053349046, −19.07367789512303382941950026971, −17.701173843557696199256560547, −16.846429135593760926140013288233, −16.310425703823513664191062967288, −14.24419414149446511355845541826, −12.83383046192083464683089004674, −12.35501380061179229762125591345, −11.1199647936006177957633584471, −10.37119513306485497573011941021, −8.51639860667287809227438061836, −7.92058856658010159222503687683, −6.78239817732928521845494860511, −4.9442678608203448282927602471, −3.5042379039349103788867689428, −1.65682771697294753131997464842, −0.37447478117766123975760516529, 2.4527635090920273272985065182, 4.42733316504479419297328627148, 5.49468821112329854676141885110, 6.77854369438199060154691614506, 7.87238958915608308491334183537, 9.30187453609510119108907939557, 10.059894267926856996491412398911, 11.30117390851210243471282787793, 12.07587978047978484367049671286, 14.40962139690384127287076767023, 15.19443436907920391135172070678, 15.657972912400030441353802818016, 16.976856094942592408603675878035, 17.84672275213495385192685116666, 18.8166885033829148088074279085, 19.84784513292343636737992384609, 21.17694315830742618306463286876, 22.38577107419725734367040175054, 23.113831906359103788000493667552, 24.134517880372179801084124191421, 25.46243419398967150159235343923, 26.20393965075029578074748832129, 27.35999486941517404460220205562, 27.6978855097921510905111829410, 28.66465188832933834077793812348

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