Properties

Degree 1
Conductor $ 29 \cdot 139 $
Sign $0.996 + 0.0791i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.699 − 0.715i)2-s + (−0.313 − 0.949i)3-s + (−0.0227 + 0.999i)4-s + (0.829 − 0.557i)5-s + (−0.460 + 0.887i)6-s + (0.538 − 0.842i)7-s + (0.730 − 0.682i)8-s + (−0.803 + 0.595i)9-s + (−0.979 − 0.203i)10-s + (−0.761 + 0.648i)11-s + (0.956 − 0.291i)12-s + (−0.746 − 0.665i)13-s + (−0.979 + 0.203i)14-s + (−0.789 − 0.613i)15-s + (−0.998 − 0.0455i)16-s + (−0.907 − 0.419i)17-s + ⋯
L(s,χ)  = 1  + (−0.699 − 0.715i)2-s + (−0.313 − 0.949i)3-s + (−0.0227 + 0.999i)4-s + (0.829 − 0.557i)5-s + (−0.460 + 0.887i)6-s + (0.538 − 0.842i)7-s + (0.730 − 0.682i)8-s + (−0.803 + 0.595i)9-s + (−0.979 − 0.203i)10-s + (−0.761 + 0.648i)11-s + (0.956 − 0.291i)12-s + (−0.746 − 0.665i)13-s + (−0.979 + 0.203i)14-s + (−0.789 − 0.613i)15-s + (−0.998 − 0.0455i)16-s + (−0.907 − 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.996 + 0.0791i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.996 + 0.0791i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4031\)    =    \(29 \cdot 139\)
\( \varepsilon \)  =  $0.996 + 0.0791i$
motivic weight  =  \(0\)
character  :  $\chi_{4031} (945, \cdot )$
Sato-Tate  :  $\mu(276)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4031,\ (0:\ ),\ 0.996 + 0.0791i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.3206614261 + 0.01271617927i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.3206614261 + 0.01271617927i\)
\(L(\chi,1)\)  \(\approx\)  \(0.4776495252 - 0.4025214492i\)
\(L(1,\chi)\)  \(\approx\)  \(0.4776495252 - 0.4025214492i\)

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

−18.08643993899030170705835182708, −17.71695307702937594720641170183, −17.25467175599396693536091786572, −16.41154107536244657894003193947, −15.64124818163598394952502047804, −15.31694520423526083385361246307, −14.59353101270190034768530111775, −13.96078183654998095288114988777, −13.32651271595930884954313452763, −11.96210879117934799673912436645, −11.304760797639591529696548854040, −10.68044985648278077374863227238, −10.147552853625698856995768999961, −9.30438421660114635448974973782, −8.91651623570923404113240408193, −8.21657850257248730168497182618, −7.13229517147508808843431005401, −6.45802603005142785704317331729, −5.72626809139672006078915568542, −5.15718432404471812856437370799, −4.65940769964349158110425474811, −3.30587074275115726894542964878, −2.36169114736764261343761977055, −1.765389132598904041809377945071, −0.133611995359255459977659187859, 0.87180720102705183008746491665, 1.639965603047780151648677169833, 2.28655723127871587718512672071, 2.931789850164243097358501420897, 4.40453275356324595700350601538, 4.834185118146800549563640140426, 5.83414806946226938108468431131, 6.76493059812367238247968626314, 7.47106572133917739892881929023, 8.04812783464094381221812873428, 8.61324107258093115789863409220, 9.66424842314156825442983884699, 10.3383237839578923633763131070, 10.66361713382539691520577517387, 11.77731723128947278168222741668, 12.23384608781421289830588208269, 12.95554238982047082975466101288, 13.45679735324984520541747859905, 14.00040003894532767785566598819, 15.022441551026930028294919115085, 16.16968336087500222664276694973, 16.901394417771826451988977595947, 17.236508609469563496193895852279, 17.96903592129832777598740459365, 18.20822332846738857848556698784

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