↑

Properties

Label 1-127-127.64-r0-0-0
Degree $1$
Conductor $127$
Sign $0.745 + 0.667i$
Analytic cond. $0.589785$
Root an. cond. $0.589785$
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  + (0.623 − 0.781i)2-s + (−0.900 + 0.433i)3-s + (−0.222 − 0.974i)4-s + (−0.222 + 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.222 + 0.974i)7-s + (−0.900 − 0.433i)8-s + (0.623 − 0.781i)9-s + (0.623 + 0.781i)10-s + (0.623 + 0.781i)11-s + (0.623 + 0.781i)12-s + (−0.222 + 0.974i)13-s + (0.623 + 0.781i)14-s + (−0.222 − 0.974i)15-s + (−0.900 + 0.433i)16-s + (−0.222 + 0.974i)17-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.900 + 0.433i)3-s + (−0.222 − 0.974i)4-s + (−0.222 + 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.222 + 0.974i)7-s + (−0.900 − 0.433i)8-s + (0.623 − 0.781i)9-s + (0.623 + 0.781i)10-s + (0.623 + 0.781i)11-s + (0.623 + 0.781i)12-s + (−0.222 + 0.974i)13-s + (0.623 + 0.781i)14-s + (−0.222 − 0.974i)15-s + (−0.900 + 0.433i)16-s + (−0.222 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(127\)
Sign: $0.745 + 0.667i$
Analytic conductor: \(0.589785\)
Root analytic conductor: \(0.589785\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{127} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 127,\ (0:\ ),\ 0.745 + 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8630801635 + 0.3299011365i\)
\(L(\frac12)\) \(\approx\) \(0.8630801635 + 0.3299011365i\)
\(L(1)\) \(\approx\) \(0.9735820129 + 0.04575711985i\)
\(L(1)\) \(\approx\) \(0.9735820129 + 0.04575711985i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad127 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 + T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (-0.900 + 0.433i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (-0.900 + 0.433i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (0.623 - 0.781i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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

−28.98185660901154710224719105268, −27.40589741770771202931385893764, −26.956208891856312465279009207933, −25.195859281222338682636753166891, −24.543397646080498263638716012623, −23.722168697837935071048025462210, −22.877357726577145455654416533711, −22.082228281353108346518525177321, −20.73818501784167401362664980636, −19.67439748321110150191313984620, −18.05049601036691158081058912373, −17.06392122741947618397817835855, −16.49336251482984370450466942085, −15.59441997106065935101687672079, −13.81954080606725618397284136835, −13.204980696679289505352238000744, −12.138879059759900750794410263475, −11.17712463889650835217812319845, −9.39123381218418254740126157688, −7.86968009998398493717862322573, −7.06380969697436539853438419864, −5.69821466023046821698775768014, −4.86438760852653916068893332790, −3.52738177496301326376802933376, −0.82086167860683795265684895494, 1.979965370289076860183858043109, 3.510510662186980123787674825366, 4.65490920947506982666268045918, 5.99034067195366672243405396031, 6.83760392446706849417164152409, 9.25922193817436238260520928974, 10.09831017720623666000603008582, 11.37294898645138898402789632133, 11.841832172573045004830053792242, 12.98943976228184432174894852075, 14.79767647451142595380298544764, 15.03211823631765798997180650806, 16.515945911239284100766763644818, 18.06341373358634328248139611637, 18.699136203401048910082478568559, 19.844276808542861571189410936960, 21.249180003458085321019558037839, 22.10118206250146482984843503213, 22.5087175413493382310382368348, 23.533155337010120365840466272652, 24.617140642085709139864974468200, 26.20670871111944258076051966330, 27.24738645024806572278471348491, 28.29136173457463506694598162301, 28.788407852786400475513163891619

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