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Properties

Label 1-187-187.90-r0-0-0
Degree $1$
Conductor $187$
Sign $0.480 + 0.877i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.987 + 0.156i)2-s + (0.760 + 0.649i)3-s + (0.951 − 0.309i)4-s + (0.972 − 0.233i)5-s + (−0.852 − 0.522i)6-s + (0.649 + 0.760i)7-s + (−0.891 + 0.453i)8-s + (0.156 + 0.987i)9-s + (−0.923 + 0.382i)10-s + (0.923 + 0.382i)12-s + (−0.587 + 0.809i)13-s + (−0.760 − 0.649i)14-s + (0.891 + 0.453i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (−0.453 − 0.891i)19-s + ⋯
L(s)  = 1  + (−0.987 + 0.156i)2-s + (0.760 + 0.649i)3-s + (0.951 − 0.309i)4-s + (0.972 − 0.233i)5-s + (−0.852 − 0.522i)6-s + (0.649 + 0.760i)7-s + (−0.891 + 0.453i)8-s + (0.156 + 0.987i)9-s + (−0.923 + 0.382i)10-s + (0.923 + 0.382i)12-s + (−0.587 + 0.809i)13-s + (−0.760 − 0.649i)14-s + (0.891 + 0.453i)15-s + (0.809 − 0.587i)16-s + (−0.309 − 0.951i)18-s + (−0.453 − 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.480 + 0.877i$
Analytic conductor: \(0.868424\)
Root analytic conductor: \(0.868424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (0:\ ),\ 0.480 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9953968545 + 0.5900013940i\)
\(L(\frac12)\) \(\approx\) \(0.9953968545 + 0.5900013940i\)
\(L(1)\) \(\approx\) \(0.9882436244 + 0.3476944025i\)
\(L(1)\) \(\approx\) \(0.9882436244 + 0.3476944025i\)

Euler product

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

−26.91915508156600886638574122904, −25.98287192997203523224367481321, −25.300049646645980800834888178651, −24.54188227594611960357538934499, −23.571388021838120709084113474522, −21.90297702989165363851919533447, −20.81730603260180242818084220347, −20.25615403828888114814737941132, −19.25263769410898108246661049372, −18.17678691557290971662427608002, −17.61937789797084718683992567190, −16.7340500583325557636030471051, −15.11562560062578531678031091763, −14.29662284362080590138826185508, −13.2031324225145025279493332873, −12.140970071308379976217577204000, −10.6976282409376776422755058651, −9.91978062013667234002729231809, −8.82676834439778258585634043616, −7.75073796189558307119046148205, −7.023246435543675472900477292062, −5.70577470631053803066426902520, −3.53415586590409855132566242530, −2.23776831473312711419226860515, −1.31265946163940032315746082763, 1.91385071510417470817583470259, 2.56158238633971116673928246649, 4.63932750057175231248513312271, 5.80593338548285968582583301006, 7.215708540728890543341879690559, 8.61544926331205973879057448153, 9.05057128763020573403956057218, 10.04403854124167843906378366753, 11.02175352877199591005294561882, 12.361489522537961325713777983289, 13.95649715667397547569755227799, 14.72969200574377138523398051458, 15.69290830813815222334220644500, 16.745409916028749135010757683, 17.60778988679001774805514761014, 18.65575916291415966670628961367, 19.56501585039437553303651329053, 20.69374127168758844677269480560, 21.29581381680828794994878492019, 22.09149537398056666348866205680, 24.20042911234916507570153578441, 24.67129384043054577627649875191, 25.672812174616061245627243817043, 26.26607823690223430878982125327, 27.27466737602859374343502833071

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