Properties

Degree $1$
Conductor $137$
Sign $0.676 - 0.736i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)6-s + i·7-s i·8-s + i·9-s + (0.707 + 0.707i)10-s + i·11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s − 14-s − 15-s + 16-s i·17-s + ⋯
L(s,χ)  = 1  + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (0.707 − 0.707i)5-s + (0.707 − 0.707i)6-s + i·7-s i·8-s + i·9-s + (0.707 + 0.707i)10-s + i·11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s − 14-s − 15-s + 16-s i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.676 - 0.736i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.676 - 0.736i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.676 - 0.736i$
Motivic weight: \(0\)
Character: $\chi_{137} (10, \cdot )$
Sato-Tate group: $\mu(8)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ 0.676 - 0.736i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.9119370487 - 0.4005948291i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.9119370487 - 0.4005948291i\)
\(L(\chi,1)\) \(\approx\) \(0.8092024772 + 0.07694016393i\)
\(L(1,\chi)\) \(\approx\) \(0.8092024772 + 0.07694016393i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.779711046146746846893420075171, −27.120782195064064870776519199567, −26.91046454126885841505345023656, −25.90275305643127499215128033293, −23.960494120928674803162237076249, −23.09587984538299670804262908948, −22.07035948736445773960382212258, −21.517679115407448506757121970886, −20.62086173920731457731148271122, −19.32085532034278552963632377861, −18.334274425131357552667400249638, −17.171997210653469882419156203334, −16.63972846793003444935288002526, −14.68246538073325572555270285617, −13.9980221321239370907789047697, −12.69645844130938457167094768760, −11.40797535916881160614427968693, −10.51320887693056163920032436009, −10.05149458080990959543307943090, −8.70367224601095065439318520736, −6.75871590430226684457956720526, −5.49275110418502553929323853370, −4.17894605489735166398395531195, −3.12425007099588147151476724126, −1.28676298193048648106274123780, 0.475943380719193722732750755034, 2.253941653898468531294932983084, 4.98024440314503913983431770091, 5.33555760385583627382526620155, 6.63450856037186792099896649335, 7.66701167979688978171638729955, 8.99174399334754675594553977465, 9.93182046813709648144355997700, 11.86056877003837135814387483612, 12.78919875283579627359448795731, 13.5148952317878959655484764147, 14.964540645313459567165416323675, 15.95616638568292425342015130542, 17.171432498591053248503616499208, 17.713859928416021725450342577624, 18.52742971491378377804465422191, 19.88576289174484638967430018687, 21.54123930085933173324475280514, 22.35194418852111558988255517257, 23.28434044845156125551208311349, 24.411894150058172655225255464408, 25.03827195524242809969339697523, 25.5539820532886363716476938106, 27.27733315278915335104771196424, 28.16434193123657491631853222981

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