Properties

Degree 1
Conductor $ 11 \cdot 17 $
Sign $0.480 + 0.877i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(187\)    =    \(11 \cdot 17\)
\( \varepsilon \)  =  $0.480 + 0.877i$
motivic weight  =  \(0\)
character  :  $\chi_{187} (90, \cdot )$
Sato-Tate  :  $\mu(80)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 187,\ (0:\ ),\ 0.480 + 0.877i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.9953968545 + 0.5900013940i$
$L(\frac12,\chi)$  $\approx$  $0.9953968545 + 0.5900013940i$
$L(\chi,1)$  $\approx$  0.9882436244 + 0.3476944025i
$L(1,\chi)$  $\approx$  0.9882436244 + 0.3476944025i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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