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Properties

Label 1-137-137.36-r0-0-0
Degree $1$
Conductor $137$
Sign $0.924 + 0.380i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.924 + 0.380i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.924 + 0.380i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9998934983 + 0.1975461597i\)
\(L(\frac12)\) \(\approx\) \(0.9998934983 + 0.1975461597i\)
\(L(1)\) \(\approx\) \(0.9817378420 + 0.07536045959i\)
\(L(1)\) \(\approx\) \(0.9817378420 + 0.07536045959i\)

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

−28.32078737178788797770250702618, −27.61018033937361342202205741204, −25.935312351393173834266119971465, −25.400742807950294882689405363214, −24.80355221680464717605863266647, −23.86966420404640354292613117800, −22.84871662070236777430987502759, −21.00008545745091955143462218561, −20.315121805572186301782369863883, −19.07570627561382751970486331077, −18.11434200782528408638995581142, −17.63482556645223657988397471944, −16.29083056206016544630559326431, −15.12955628308284920401779225420, −14.208607078132138411931093885057, −13.026391403074239099296281233638, −12.01803246374807629903696984207, −10.29245854293738972637927640697, −8.88013021917422276443341172353, −8.62894419021820084364536154078, −7.18455849934275084766310737328, −6.072085832461558748507177001813, −4.989977055160969719212714694152, −2.41229882200272799739454102682, −1.33748372354046007387347848263, 1.72981056859313001718990935433, 3.23786421257657027701816659306, 4.03133775925398762819939088450, 6.05597736010578816749857736009, 7.68735808629197046003242095562, 8.66960637868393747238289030966, 9.88820770656453033158046253888, 10.65020168873722867013437797951, 11.29713888717167285819875499068, 13.36342055940206829288157342622, 13.94655097909974141415984477404, 15.34114421581632623547625658550, 16.69548667184931011805396637239, 17.27631119303580193783937908057, 18.75315005901913197266989488291, 19.39211327435526453558257901093, 20.813385931956955604615519470573, 21.18032741811745007864779300222, 22.116694931344569443573225578539, 23.38283748227581264984983946462, 25.145024914924934375588775509150, 26.03892387762117692270027304651, 26.527968040372361046441455174464, 27.43831199830491234186307287628, 28.38862956534548518950774906853

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