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Properties

Label 1-137-137.10-r1-0-0
Degree $1$
Conductor $137$
Sign $0.676 - 0.736i$
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  + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.676 - 0.736i$
Analytic conductor: \(14.7226\)
Root analytic conductor: \(14.7226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (1:\ ),\ 0.676 - 0.736i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad137 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + iT \)
11 \( 1 + iT \)
13 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 - iT \)
19 \( 1 - iT \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−28.779711046146746846893420075171, −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