Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $-0.685 + 0.728i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.998 + 0.0475i)2-s + (0.771 − 0.636i)3-s + (0.995 − 0.0950i)4-s + (0.142 + 0.989i)5-s + (−0.739 + 0.672i)6-s + (−0.992 − 0.118i)7-s + (−0.989 + 0.142i)8-s + (0.189 − 0.981i)9-s + (−0.189 − 0.981i)10-s + (0.618 − 0.786i)11-s + (0.707 − 0.707i)12-s + (0.997 + 0.0713i)14-s + (0.739 + 0.672i)15-s + (0.981 − 0.189i)16-s + (−0.998 − 0.0475i)17-s + (−0.142 + 0.989i)18-s + ⋯
L(s,χ)  = 1  + (−0.998 + 0.0475i)2-s + (0.771 − 0.636i)3-s + (0.995 − 0.0950i)4-s + (0.142 + 0.989i)5-s + (−0.739 + 0.672i)6-s + (−0.992 − 0.118i)7-s + (−0.989 + 0.142i)8-s + (0.189 − 0.981i)9-s + (−0.189 − 0.981i)10-s + (0.618 − 0.786i)11-s + (0.707 − 0.707i)12-s + (0.997 + 0.0713i)14-s + (0.739 + 0.672i)15-s + (0.981 − 0.189i)16-s + (−0.998 − 0.0475i)17-s + (−0.142 + 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.685 + 0.728i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.685 + 0.728i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $-0.685 + 0.728i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (24, \cdot )$
Sato-Tate  :  $\mu(264)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1157,\ (0:\ ),\ -0.685 + 0.728i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1118505053 + 0.2588575030i$
$L(\frac12,\chi)$  $\approx$  $0.1118505053 + 0.2588575030i$
$L(\chi,1)$  $\approx$  0.6523333215 + 0.01106916285i
$L(1,\chi)$  $\approx$  0.6523333215 + 0.01106916285i

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

−20.52873003611606180070212574364, −20.23320120884789331196608166354, −19.50083023946565666637170411098, −19.04655976748795787902623873317, −17.67266925202809412855060756553, −17.19641537105162355649480103026, −16.26200484619839605963171833009, −15.74235737990256138648599616870, −15.17291747113763650028087573990, −14.04183818359588821490203838667, −13.00703624751384864890220421832, −12.43829523911948275569896744713, −11.371652559432247963608938390275, −10.346636831826226445081695045348, −9.62590867496166624730197785702, −9.11658352380954476402408851624, −8.57718705829121803574041012891, −7.57009534553708426688844552236, −6.6604180968716039626314730276, −5.674392390130039195089504467994, −4.38912550004067280358503322366, −3.688501219542731232881753908088, −2.37189631077361162754570790003, −1.81827522107481765027757673463, −0.13230148455243022048441877508, 1.39565586668846530520831008162, 2.38865540762402627260988939966, 3.11516263884135719293821654887, 3.8816868264605667063354623209, 6.10485958646706574633749789893, 6.4371709784784000246741164885, 7.08153192303815328255463942632, 8.07496256681873032507362680896, 8.80262517223321603615321890810, 9.5743978585580021877236447898, 10.33276913301693630939915274240, 11.162680082828540206138041852590, 12.075573323551791354232008455919, 12.98337492410834760808676380739, 13.881682165242820951608234918859, 14.65787652297926893295586002495, 15.32341645534395760288356289861, 16.274090066033957828754219725501, 16.99610647435170856565739039642, 18.14044333350081282301180096164, 18.42247589068988340958676789529, 19.27612431759897103994736281463, 19.724350167622193957136882713163, 20.34870384729471602778953540243, 21.570744963533355360603818233219

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