↑

Properties

Label 1-139-139.16-r0-0-0
Degree $1$
Conductor $139$
Sign $0.00655 + 0.999i$
Analytic cond. $0.645513$
Root an. cond. $0.645513$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.983 + 0.181i)2-s + (0.377 + 0.926i)3-s + (0.934 + 0.356i)4-s + (−0.998 + 0.0455i)5-s + (0.203 + 0.979i)6-s + (−0.949 + 0.313i)7-s + (0.854 + 0.519i)8-s + (−0.715 + 0.699i)9-s + (−0.990 − 0.136i)10-s + (0.291 + 0.956i)11-s + (0.0227 + 0.999i)12-s + (0.613 − 0.789i)13-s + (−0.990 + 0.136i)14-s + (−0.419 − 0.907i)15-s + (0.746 + 0.665i)16-s + (0.803 + 0.595i)17-s + ⋯
L(s)  = 1  + (0.983 + 0.181i)2-s + (0.377 + 0.926i)3-s + (0.934 + 0.356i)4-s + (−0.998 + 0.0455i)5-s + (0.203 + 0.979i)6-s + (−0.949 + 0.313i)7-s + (0.854 + 0.519i)8-s + (−0.715 + 0.699i)9-s + (−0.990 − 0.136i)10-s + (0.291 + 0.956i)11-s + (0.0227 + 0.999i)12-s + (0.613 − 0.789i)13-s + (−0.990 + 0.136i)14-s + (−0.419 − 0.907i)15-s + (0.746 + 0.665i)16-s + (0.803 + 0.595i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00655 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00655 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(139\)
Sign: $0.00655 + 0.999i$
Analytic conductor: \(0.645513\)
Root analytic conductor: \(0.645513\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{139} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 139,\ (0:\ ),\ 0.00655 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.247981512 + 1.239829976i\)
\(L(\frac12)\) \(\approx\) \(1.247981512 + 1.239829976i\)
\(L(1)\) \(\approx\) \(1.445539708 + 0.7946152846i\)
\(L(1)\) \(\approx\) \(1.445539708 + 0.7946152846i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad139 \( 1 \)
good2 \( 1 + (0.983 + 0.181i)T \)
3 \( 1 + (0.377 + 0.926i)T \)
5 \( 1 + (-0.998 + 0.0455i)T \)
7 \( 1 + (-0.949 + 0.313i)T \)
11 \( 1 + (0.291 + 0.956i)T \)
13 \( 1 + (0.613 - 0.789i)T \)
17 \( 1 + (0.803 + 0.595i)T \)
19 \( 1 + (0.113 - 0.993i)T \)
23 \( 1 + (0.203 - 0.979i)T \)
29 \( 1 + (-0.158 - 0.987i)T \)
31 \( 1 + (-0.715 - 0.699i)T \)
37 \( 1 + (-0.419 + 0.907i)T \)
41 \( 1 + (0.898 - 0.439i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.538 - 0.842i)T \)
53 \( 1 + (-0.648 + 0.761i)T \)
59 \( 1 + (-0.775 - 0.631i)T \)
61 \( 1 + (0.898 + 0.439i)T \)
67 \( 1 + (-0.974 + 0.225i)T \)
71 \( 1 + (0.538 + 0.842i)T \)
73 \( 1 + (-0.877 + 0.480i)T \)
79 \( 1 + (-0.0682 - 0.997i)T \)
83 \( 1 + (-0.974 - 0.225i)T \)
89 \( 1 + (0.995 + 0.0909i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.62100648859893730290991607432, −27.15870422726684286766307208568, −25.86789601807502217136547839521, −25.0422562882731649097718313726, −23.86952535445992232853051382889, −23.38700218083252554379225239788, −22.53226178856615866094184126966, −21.12456697641010163071416927348, −20.04697163460627519539455915557, −19.2685383044044074397698967516, −18.67856312898686943280880250295, −16.55502082945134346214392599739, −15.97991021856768523238787685914, −14.48503538773360316566128769078, −13.78905054309526683158583386656, −12.68934678550300189146684671443, −11.912434351456090052986998446150, −10.92669295362198998439692335179, −9.12851443780636418081954508095, −7.66213876042140265138860731836, −6.80062090623904413351960435372, −5.69017910456386189473197605223, −3.712205996828637726211629654624, −3.22761127204462442493000075863, −1.278698357461548804874876640577, 2.73381600334405452569580289226, 3.66645609464581338901409152819, 4.60329128011053973253087544235, 5.960296645095358174308496314863, 7.35322510092435782842481726903, 8.53625065763568559895382854260, 10.01767131614030742271558901064, 11.12487435692469959507635622257, 12.288263606955748749145103768842, 13.225148448412274753235212247, 14.726075906241512066119902244526, 15.32336261999510028301206689028, 16.0266969886141919967596562374, 17.03610257066484588369028154255, 19.042547055195915921895956696639, 20.037286567637242675233243665872, 20.62415230982927897729038221060, 21.96919295834326512990858727560, 22.69087943217224357235322332353, 23.31890857764043812745594221574, 24.78092848843624153459275196139, 25.74628758019331246657019707055, 26.3728436152270068937079358622, 27.898177310074762806142382222313, 28.39747682261057907488010241163

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