↑

Properties

Label 1-187-187.40-r0-0-0
Degree $1$
Conductor $187$
Sign $0.820 + 0.571i$
Analytic cond. $0.868424$
Root an. cond. $0.868424$
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  + (0.453 − 0.891i)2-s + (−0.972 − 0.233i)3-s + (−0.587 − 0.809i)4-s + (−0.996 + 0.0784i)5-s + (−0.649 + 0.760i)6-s + (0.233 + 0.972i)7-s + (−0.987 + 0.156i)8-s + (0.891 + 0.453i)9-s + (−0.382 + 0.923i)10-s + (0.382 + 0.923i)12-s + (−0.951 + 0.309i)13-s + (0.972 + 0.233i)14-s + (0.987 + 0.156i)15-s + (−0.309 + 0.951i)16-s + (0.809 − 0.587i)18-s + (0.156 + 0.987i)19-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)2-s + (−0.972 − 0.233i)3-s + (−0.587 − 0.809i)4-s + (−0.996 + 0.0784i)5-s + (−0.649 + 0.760i)6-s + (0.233 + 0.972i)7-s + (−0.987 + 0.156i)8-s + (0.891 + 0.453i)9-s + (−0.382 + 0.923i)10-s + (0.382 + 0.923i)12-s + (−0.951 + 0.309i)13-s + (0.972 + 0.233i)14-s + (0.987 + 0.156i)15-s + (−0.309 + 0.951i)16-s + (0.809 − 0.587i)18-s + (0.156 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ 0.820 + 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4910088406 + 0.1540865629i\)
\(L(\frac12)\) \(\approx\) \(0.4910088406 + 0.1540865629i\)
\(L(1)\) \(\approx\) \(0.6550564568 - 0.1757529656i\)
\(L(1)\) \(\approx\) \(0.6550564568 - 0.1757529656i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.453 - 0.891i)T \)
3 \( 1 + (-0.972 - 0.233i)T \)
5 \( 1 + (-0.996 + 0.0784i)T \)
7 \( 1 + (0.233 + 0.972i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (0.156 + 0.987i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (0.852 + 0.522i)T \)
31 \( 1 + (-0.649 - 0.760i)T \)
37 \( 1 + (-0.522 + 0.852i)T \)
41 \( 1 + (-0.852 + 0.522i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.453 + 0.891i)T \)
59 \( 1 + (0.156 - 0.987i)T \)
61 \( 1 + (0.760 + 0.649i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.0784 - 0.996i)T \)
73 \( 1 + (-0.852 - 0.522i)T \)
79 \( 1 + (-0.0784 + 0.996i)T \)
83 \( 1 + (0.891 - 0.453i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.760 + 0.649i)T \)
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

−26.96703203847282929805756816358, −26.44936063082944062884576259578, −24.82931192001884782993730193401, −23.95323531142332975648988972720, −23.35716886125794007855854944400, −22.63407377016728555183875727337, −21.71801812508543566848976538216, −20.543430767125367125281998430, −19.31293361285982137021873497083, −17.90413038767778811272886226994, −17.14980718138798373461702772834, −16.3632782454051591860556397759, −15.47939990169494130311721495069, −14.60665769166156921018068411655, −13.22890553272310323380549645798, −12.27494564315240647370352583667, −11.34743350118839807152539519935, −10.18225979584539710823156644495, −8.65017079047886843441235060063, −7.27582611248921139277920255401, −6.89357729688513985917841973097, −5.17071783974029475398140887852, −4.556053729502093112400150071084, −3.42726546317368671137956301319, −0.415914332873584640745752239459, 1.54239232810493524800287108520, 3.066609578917933203826161151788, 4.5405795265148203928170022109, 5.30787480281353272004175999558, 6.601093927225486067273144571392, 8.03186183014181442381161006146, 9.46297445199666190567374045455, 10.665320250848356379837092703340, 11.69848613262513911795747420737, 12.0780260979052378007020057055, 13.00241520784243697991766187045, 14.57330567370428790860715377310, 15.377164712424803896639890125, 16.57566401917613763556410686983, 17.89530142438366408889794841435, 18.811132209030125121425176318146, 19.347124739526295566992964059126, 20.67395902173432931947779466440, 21.77930480066287487264428153550, 22.38543290049879021138328633487, 23.325344184431625476653489075398, 24.04494753380342650970845415322, 24.95425798264520455025342439295, 26.979398403363979545126652362251, 27.43738653041673168773137814479

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