Properties

Degree 1
Conductor 31
Sign $0.999 + 0.00222i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.809 − 0.587i)2-s + (0.913 + 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (−0.978 − 0.207i)11-s + (−0.104 + 0.994i)12-s + (−0.104 − 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (−0.978 + 0.207i)17-s + ⋯
L(s,χ)  = 1  + (−0.809 − 0.587i)2-s + (0.913 + 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (−0.978 − 0.207i)11-s + (−0.104 + 0.994i)12-s + (−0.104 − 0.994i)13-s + (−0.978 + 0.207i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (−0.978 + 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.999 + 0.00222i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.999 + 0.00222i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $0.999 + 0.00222i$
motivic weight  =  \(0\)
character  :  $\chi_{31} (9, \cdot )$
Sato-Tate  :  $\mu(15)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 31,\ (0:\ ),\ 0.999 + 0.00222i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6554674089 + 0.0007280021081i$
$L(\frac12,\chi)$  $\approx$  $0.6554674089 + 0.0007280021081i$
$L(\chi,1)$  $\approx$  0.8287949537 + 0.02697815852i
$L(1,\chi)$  $\approx$  0.8287949537 + 0.02697815852i

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

−36.62180951710503200400478120794, −35.786189018350751291183895231476, −34.67655966463691943904815805798, −33.26939857420932421714620449708, −31.76763754141801505547696795324, −31.07952168970990083661404975181, −29.06406048176813209844528384434, −27.97702394928114803206234035563, −26.70828708802156378297478535269, −25.58163869110194213736425930101, −24.28702817211852782547577275791, −23.85098500147151386574802055835, −21.11981847472883910359385090190, −19.96636075835652272066513364295, −18.8394497802142786248619011841, −17.66329959862695496636846770262, −15.87892828465175943007024104820, −14.99499092204154228890446384519, −13.323424700965857289848156755314, −11.51996630368864209215613312180, −9.266506068824407895498415392169, −8.4473590942751644470298435269, −7.19028270903748450743965441993, −4.95271728153537707525623277335, −1.98829132601605799494657696959, 2.53375549830066134190223605277, 3.9955241988163664897829608375, 7.458957902275047094233536329113, 8.30135624286073415779836885454, 10.25055912822505416674286265980, 10.95326656741069251161491819798, 13.08182979585236854245239050099, 14.72799377082321837998853508372, 16.04162315963009589522886130447, 17.846347702779190438834596238698, 19.02714900809254327906793264300, 20.20914145445735977555706398718, 21.107338053179973782265572861583, 22.64515749542138313094737326260, 24.63841745043031974679881870131, 26.22427527340578477573072511272, 26.73063218536350799887946677506, 27.75300249129972543283861817814, 29.61031998397484575705064615932, 30.59516150190442147597336986489, 31.460538093618163556896661032687, 33.34554613893120470068398033851, 34.50500890708873703500932080590, 35.93855148924096073400996994877, 37.11299394054410036693969436955

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