Properties

Degree 1
Conductor $ 13 \cdot 89 $
Sign $0.147 - 0.989i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.458 − 0.888i)2-s + (−0.986 − 0.165i)3-s + (−0.580 + 0.814i)4-s + (0.142 − 0.989i)5-s + (0.304 + 0.952i)6-s + (0.919 + 0.393i)7-s + (0.989 + 0.142i)8-s + (0.945 + 0.327i)9-s + (−0.945 + 0.327i)10-s + (−0.371 + 0.928i)11-s + (0.707 − 0.707i)12-s + (−0.0713 − 0.997i)14-s + (−0.304 + 0.952i)15-s + (−0.327 − 0.945i)16-s + (−0.458 + 0.888i)17-s + (−0.142 − 0.989i)18-s + ⋯
L(s,χ)  = 1  + (−0.458 − 0.888i)2-s + (−0.986 − 0.165i)3-s + (−0.580 + 0.814i)4-s + (0.142 − 0.989i)5-s + (0.304 + 0.952i)6-s + (0.919 + 0.393i)7-s + (0.989 + 0.142i)8-s + (0.945 + 0.327i)9-s + (−0.945 + 0.327i)10-s + (−0.371 + 0.928i)11-s + (0.707 − 0.707i)12-s + (−0.0713 − 0.997i)14-s + (−0.304 + 0.952i)15-s + (−0.327 − 0.945i)16-s + (−0.458 + 0.888i)17-s + (−0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.147 - 0.989i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.147 - 0.989i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1157\)    =    \(13 \cdot 89\)
\( \varepsilon \)  =  $0.147 - 0.989i$
motivic weight  =  \(0\)
character  :  $\chi_{1157} (362, \cdot )$
Sato-Tate  :  $\mu(264)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 1157,\ (0:\ ),\ 0.147 - 0.989i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6266385788 - 0.5402260012i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6266385788 - 0.5402260012i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6013193790 - 0.3181964744i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6013193790 - 0.3181964744i\)

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

−21.6404025616629244108545462766, −20.89920533068548867831995380349, −19.62656552116753201121837470849, −18.6538143135637469666683840979, −18.225565630133871627477709772452, −17.65700544375326729395249146484, −16.77425196576853256726725542491, −16.258507577076490011421514031847, −15.26674933266542436222006142449, −14.69964663641073723955909668342, −13.81770058501425725295641351966, −13.13441806569120181460117279820, −11.58328923481701600261001980569, −11.0889948199636996156541573179, −10.44633881391379091394958350481, −9.68048130396925023914280578792, −8.57813906099354645218696825700, −7.536014849197652569529016841350, −7.07226939894294530553114425554, −6.057901443152272933443668571081, −5.49410273837070509511788370453, −4.592412968316856178425967385262, −3.57164995005947156716773998933, −1.95843986297638997044994167265, −0.779873592763754296564720451708, 0.69451663635722722342679994351, 1.72871670255333913589764700063, 2.33167414194086736488799891323, 4.14942023456726939015789070574, 4.72524173707084492634470085213, 5.315508288296859924817940384088, 6.62074794410074003289654859310, 7.68632348609012046241792844048, 8.48642575943494061087638310328, 9.20018319605155509256439628341, 10.27717753772561919332966844090, 10.82158614177356705069562305459, 11.76412164839551111622565593297, 12.30953872629444078198682753742, 12.93493283460332386677612899163, 13.64285878482852947883863405860, 15.07786986703748536478170207731, 15.83760343347879459365000056458, 17.004944353550114297686391329807, 17.32179652434041860261899792764, 17.877857935516124996143728775341, 18.72745324994963936423963502542, 19.537646772972537488188251369908, 20.55819769070457836846895859134, 20.96045735036244867534104722759

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