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Properties

Label 1-137-137.112-r0-0-0
Degree $1$
Conductor $137$
Sign $0.654 + 0.755i$
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  + (0.982 + 0.183i)2-s + (−0.798 − 0.602i)3-s + (0.932 + 0.361i)4-s + (0.183 + 0.982i)5-s + (−0.673 − 0.739i)6-s + (−0.445 + 0.895i)7-s + (0.850 + 0.526i)8-s + (0.273 + 0.961i)9-s + i·10-s + (−0.932 − 0.361i)11-s + (−0.526 − 0.850i)12-s + (0.895 + 0.445i)13-s + (−0.602 + 0.798i)14-s + (0.445 − 0.895i)15-s + (0.739 + 0.673i)16-s + (0.850 − 0.526i)17-s + ⋯
L(s)  = 1  + (0.982 + 0.183i)2-s + (−0.798 − 0.602i)3-s + (0.932 + 0.361i)4-s + (0.183 + 0.982i)5-s + (−0.673 − 0.739i)6-s + (−0.445 + 0.895i)7-s + (0.850 + 0.526i)8-s + (0.273 + 0.961i)9-s + i·10-s + (−0.932 − 0.361i)11-s + (−0.526 − 0.850i)12-s + (0.895 + 0.445i)13-s + (−0.602 + 0.798i)14-s + (0.445 − 0.895i)15-s + (0.739 + 0.673i)16-s + (0.850 − 0.526i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.654 + 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.369487842 + 0.6252743470i\)
\(L(\frac12)\) \(\approx\) \(1.369487842 + 0.6252743470i\)
\(L(1)\) \(\approx\) \(1.394925118 + 0.3352598918i\)
\(L(1)\) \(\approx\) \(1.394925118 + 0.3352598918i\)

Euler product

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

−28.4824855892946567905040402547, −27.80308327253895545701620648806, −26.282503352684985983266588699549, −25.24916983482958508355697101519, −23.88209216413253875438140433408, −23.30665702839617488933726862994, −22.58947765025976397873382524483, −21.07464832838613238065117789004, −20.87422998930147764976704851075, −19.773272523532215964668167554262, −18.0851392409425777575971807628, −16.62210215125672694177713882034, −16.287683974930800622380924874785, −15.16311836997486707908085297547, −13.72059070947201938594474361247, −12.75811554585442000767448422463, −12.015358828669283761812678794622, −10.434012762758925716277304029047, −10.12375658870210082638545681407, −8.08842234507061765970565206513, −6.44350047815675985917399181705, −5.48723870525876828950499207342, −4.47534514267685589775955740587, −3.458695836911656049102561652694, −1.229583150990967550781403442032, 2.15974115090624983767119404940, 3.253034356549975879396417057240, 5.153197663369871158961516812503, 6.055012487754869057875205102511, 6.83406402899005634890635017644, 8.071552740793517408767899674365, 10.1887836960762100490626358919, 11.31487269650161388604669590497, 12.01168128598366589298229530515, 13.29949161055081086344676702572, 13.93594508403049710745895189039, 15.52059044637624252498447858201, 16.031403858289431619133057593255, 17.52813118663580863244403879292, 18.563370886075565925716638672829, 19.32427059105597224962366410278, 21.19641122379853073141781182774, 21.7899269929055323198920486962, 22.8328635967945680951302087780, 23.38733983125280933613832563204, 24.46232851899354130003121591721, 25.52066276367056415464969737314, 26.21002686723507415546782134413, 28.02129127114082398223498387684, 28.9194089010470286747468460214

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