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Properties

Label 1-137-137.105-r0-0-0
Degree $1$
Conductor $137$
Sign $-0.144 - 0.989i$
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  + (−0.445 − 0.895i)2-s + (0.673 − 0.739i)3-s + (−0.602 + 0.798i)4-s + (0.895 + 0.445i)5-s + (−0.961 − 0.273i)6-s + (−0.932 + 0.361i)7-s + (0.982 + 0.183i)8-s + (−0.0922 − 0.995i)9-s − i·10-s + (0.602 − 0.798i)11-s + (0.183 + 0.982i)12-s + (−0.361 − 0.932i)13-s + (0.739 + 0.673i)14-s + (0.932 − 0.361i)15-s + (−0.273 − 0.961i)16-s + (0.982 − 0.183i)17-s + ⋯
L(s)  = 1  + (−0.445 − 0.895i)2-s + (0.673 − 0.739i)3-s + (−0.602 + 0.798i)4-s + (0.895 + 0.445i)5-s + (−0.961 − 0.273i)6-s + (−0.932 + 0.361i)7-s + (0.982 + 0.183i)8-s + (−0.0922 − 0.995i)9-s − i·10-s + (0.602 − 0.798i)11-s + (0.183 + 0.982i)12-s + (−0.361 − 0.932i)13-s + (0.739 + 0.673i)14-s + (0.932 − 0.361i)15-s + (−0.273 − 0.961i)16-s + (0.982 − 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ -0.144 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7152166941 - 0.8269998424i\)
\(L(\frac12)\) \(\approx\) \(0.7152166941 - 0.8269998424i\)
\(L(1)\) \(\approx\) \(0.8748129798 - 0.5945438519i\)
\(L(1)\) \(\approx\) \(0.8748129798 - 0.5945438519i\)

Euler product

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

−28.455503195709890407974023291843, −27.634804538820080162604697597552, −26.457646791399096011568113083526, −25.77680820346432158913585918189, −25.18409357623138826447567576634, −24.12303444584090786903069505456, −22.709182975237490510196022703923, −21.93749838888129111180042036846, −20.59058038602192373101006631386, −19.7053222050042226882533957858, −18.66568502466523539210481709413, −17.20348853829387170918851763025, −16.56093297774416458185135165193, −15.715855681858660191063940078483, −14.24704311636649686652254458735, −13.97114521164659197370651906334, −12.46867516109195819036508759355, −10.220226547667548345021640859035, −9.71520577769786762776297299571, −8.965178049655328871815041823109, −7.55838956719969922039083322156, −6.31841593176711646682146469966, −5.052977950523763964100092052224, −3.8379501979999974165799718297, −1.82238656461244466635319613305, 1.239527523581270282495934787696, 2.78827973542696150587588553246, 3.308688020039888198881751927332, 5.710933034539997888554856040300, 7.013791563086142709927522127586, 8.32186965971351303360094637982, 9.44463671719529498973205876540, 10.1158971668791939174386150662, 11.722122123812016433815743268599, 12.70837456516315886742948045512, 13.60801067193248648886363918866, 14.41984210981637450536428769098, 16.18515934073794471305033270124, 17.593395262077106746664805234404, 18.25182264868898599298278862107, 19.28225789111238569253065787757, 19.86699307443424023502932821070, 21.13515657485418546249288342819, 22.04422952455658736057270534002, 22.91770275084170581452717552349, 24.644005756541702933882252420789, 25.46181064289036029115302682901, 26.14776918661196302521582433245, 27.17097477734040966998921900339, 28.52969082810747604903513438541

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