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Properties

Label 1-137-137.52-r1-0-0
Degree $1$
Conductor $137$
Sign $-0.999 + 0.00443i$
Analytic cond. $14.7226$
Root an. cond. $14.7226$
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  + (−0.361 + 0.932i)2-s + (−0.486 + 0.873i)3-s + (−0.739 − 0.673i)4-s + (0.403 − 0.914i)5-s + (−0.638 − 0.769i)6-s + (0.798 − 0.602i)7-s + (0.895 − 0.445i)8-s + (−0.526 − 0.850i)9-s + (0.707 + 0.707i)10-s + (−0.673 + 0.739i)11-s + (0.948 − 0.317i)12-s + (−0.138 + 0.990i)13-s + (0.273 + 0.961i)14-s + (0.602 + 0.798i)15-s + (0.0922 + 0.995i)16-s + (−0.895 − 0.445i)17-s + ⋯
L(s)  = 1  + (−0.361 + 0.932i)2-s + (−0.486 + 0.873i)3-s + (−0.739 − 0.673i)4-s + (0.403 − 0.914i)5-s + (−0.638 − 0.769i)6-s + (0.798 − 0.602i)7-s + (0.895 − 0.445i)8-s + (−0.526 − 0.850i)9-s + (0.707 + 0.707i)10-s + (−0.673 + 0.739i)11-s + (0.948 − 0.317i)12-s + (−0.138 + 0.990i)13-s + (0.273 + 0.961i)14-s + (0.602 + 0.798i)15-s + (0.0922 + 0.995i)16-s + (−0.895 − 0.445i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $-0.999 + 0.00443i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ -0.999 + 0.00443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001563320007 + 0.7056184234i\)
\(L(\frac12)\) \(\approx\) \(0.001563320007 + 0.7056184234i\)
\(L(1)\) \(\approx\) \(0.5553995458 + 0.4189530078i\)
\(L(1)\) \(\approx\) \(0.5553995458 + 0.4189530078i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + (-0.361 + 0.932i)T \)
3 \( 1 + (-0.486 + 0.873i)T \)
5 \( 1 + (0.403 - 0.914i)T \)
7 \( 1 + (0.798 - 0.602i)T \)
11 \( 1 + (-0.673 + 0.739i)T \)
13 \( 1 + (-0.138 + 0.990i)T \)
17 \( 1 + (-0.895 - 0.445i)T \)
19 \( 1 + (0.183 + 0.982i)T \)
23 \( 1 + (-0.638 + 0.769i)T \)
29 \( 1 + (0.769 + 0.638i)T \)
31 \( 1 + (0.565 + 0.824i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (0.824 + 0.565i)T \)
47 \( 1 + (-0.228 + 0.973i)T \)
53 \( 1 + (-0.565 + 0.824i)T \)
59 \( 1 + (-0.850 + 0.526i)T \)
61 \( 1 + (-0.526 + 0.850i)T \)
67 \( 1 + (-0.990 - 0.138i)T \)
71 \( 1 + (-0.998 - 0.0461i)T \)
73 \( 1 + (-0.602 + 0.798i)T \)
79 \( 1 + (0.873 - 0.486i)T \)
83 \( 1 + (0.317 - 0.948i)T \)
89 \( 1 + (0.914 + 0.403i)T \)
97 \( 1 + (0.0461 + 0.998i)T \)
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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

−28.0363115998009985315191446604, −26.890092814682782792879407637996, −25.94433326633374299308504888563, −24.76023574484602697545866026097, −23.71302104462819399803341682145, −22.34372089414155571140165264297, −21.96682361482919352972902052614, −20.70551959380361586333413534998, −19.41926608348825960396394264777, −18.539999775805700401053261441469, −17.89561476174554262245356173393, −17.24182784019420252789785906136, −15.40502898689166787274472199299, −13.90808771809245646856004863414, −13.18308257153609404642137314242, −11.931840752758302096793925336297, −11.0537627034691979885945115655, −10.33291877914529532449274655493, −8.567495347163789531554440310018, −7.73681544494383902119103044119, −6.20993480453034539464374286616, −4.9623058466847175894345613078, −2.85256218857701039983295489889, −2.07138599312720552475561239823, −0.34410306038724725004047340748, 1.43425939381345604146672533677, 4.39348283881390315611982373942, 4.83670471994847675576332991380, 6.06644602793170626009737556921, 7.51820205709565267267524823501, 8.77914735063694438202551276322, 9.70412319941213220345037956899, 10.650998484344913221550992989098, 12.11510601805008364410906852626, 13.663423821824660988120223415401, 14.54611125681640781239771840895, 15.84925850622324394397593696456, 16.433699842086055285162900341264, 17.52968555209596361785404134619, 17.95966526325459966958453998876, 19.84521520306938852380567185237, 20.81291841302930742792004806872, 21.734556869061685092280399781957, 23.12671365908786051170891598377, 23.75188277677545264340270974955, 24.74602759234852026787099458212, 25.907721478615980555898093036440, 26.81209187986051584003905186094, 27.57267519856941105448377874760, 28.52408849398287457342439723205

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