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Properties

Label 1-137-137.15-r0-0-0
Degree $1$
Conductor $137$
Sign $0.535 - 0.844i$
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  + (−0.850 − 0.526i)2-s + (−0.932 − 0.361i)3-s + (0.445 + 0.895i)4-s + (0.850 − 0.526i)5-s + (0.602 + 0.798i)6-s + (−0.982 + 0.183i)7-s + (0.0922 − 0.995i)8-s + (0.739 + 0.673i)9-s − 10-s + (0.445 + 0.895i)11-s + (−0.0922 − 0.995i)12-s + (0.982 − 0.183i)13-s + (0.932 + 0.361i)14-s + (−0.982 + 0.183i)15-s + (−0.602 + 0.798i)16-s + (0.0922 + 0.995i)17-s + ⋯
L(s)  = 1  + (−0.850 − 0.526i)2-s + (−0.932 − 0.361i)3-s + (0.445 + 0.895i)4-s + (0.850 − 0.526i)5-s + (0.602 + 0.798i)6-s + (−0.982 + 0.183i)7-s + (0.0922 − 0.995i)8-s + (0.739 + 0.673i)9-s − 10-s + (0.445 + 0.895i)11-s + (−0.0922 − 0.995i)12-s + (0.982 − 0.183i)13-s + (0.932 + 0.361i)14-s + (−0.982 + 0.183i)15-s + (−0.602 + 0.798i)16-s + (0.0922 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(137\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(0.636225\)
Root analytic conductor: \(0.636225\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{137} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 137,\ (0:\ ),\ 0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5312668370 - 0.2921965306i\)
\(L(\frac12)\) \(\approx\) \(0.5312668370 - 0.2921965306i\)
\(L(1)\) \(\approx\) \(0.5962666407 - 0.2124698131i\)
\(L(1)\) \(\approx\) \(0.5962666407 - 0.2124698131i\)

Euler product

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

−28.91688436400557166127226581819, −27.41706739169486986312224049694, −26.8400327966674289311876870926, −25.67456851729089551736573009753, −25.02951896442874181665228111863, −23.56963669683441266165136751485, −22.85000769126862261923476958093, −21.768671368501062348536873468256, −20.68330786607399884018525759257, −19.1255540212539058263820093910, −18.40643095122764463929982275656, −17.43454520123201569726378142399, −16.43084339353006635322401056026, −15.93552561530684118576059323372, −14.45891091254770128788343668918, −13.34570975584184234332834292371, −11.61724680705104778307970820481, −10.632090486222024528632162423612, −9.82599196773433690847853364846, −8.8709574303509769235443673727, −6.940045157680151954319039451244, −6.28604720644806359104518496233, −5.40164721753733946508295323235, −3.32778300111072403844362309904, −1.20188449385769758218472623469, 1.03599267345985213891047047259, 2.37861934513343489083603924711, 4.31197010128416206173073084908, 6.08183878441042045525910530780, 6.783969228195780278843306075975, 8.44075576569986903131640909318, 9.62341756383626023469694830178, 10.4054315074259648637898582418, 11.64263506490505003097980849960, 12.84859660848389242965176323573, 13.14355412107467329729026161962, 15.48056972325165136680653171777, 16.61697343811434160548161214392, 17.25176675834090689381022871887, 18.12468155026399205476953602906, 19.10787258130830234676712472275, 20.175715885110447334708408000608, 21.35254636205033271773082754147, 22.17291097417412119739870776204, 23.210316931404269620990274774349, 24.67846572073116287469106262063, 25.43442470005697091120786563755, 26.28980991614251838499840070942, 27.96482465429000392704690930808, 28.30758497918652638451821033532

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