Properties

Degree 1
Conductor 67
Sign $-0.999 - 0.0434i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.723 + 0.690i)2-s + (0.142 − 0.989i)3-s + (0.0475 − 0.998i)4-s + (−0.415 − 0.909i)5-s + (0.580 + 0.814i)6-s + (−0.235 − 0.971i)7-s + (0.654 + 0.755i)8-s + (−0.959 − 0.281i)9-s + (0.928 + 0.371i)10-s + (−0.580 + 0.814i)11-s + (−0.981 − 0.189i)12-s + (0.327 + 0.945i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (−0.995 − 0.0950i)16-s + (0.0475 + 0.998i)17-s + ⋯
L(s,χ)  = 1  + (−0.723 + 0.690i)2-s + (0.142 − 0.989i)3-s + (0.0475 − 0.998i)4-s + (−0.415 − 0.909i)5-s + (0.580 + 0.814i)6-s + (−0.235 − 0.971i)7-s + (0.654 + 0.755i)8-s + (−0.959 − 0.281i)9-s + (0.928 + 0.371i)10-s + (−0.580 + 0.814i)11-s + (−0.981 − 0.189i)12-s + (0.327 + 0.945i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (−0.995 − 0.0950i)16-s + (0.0475 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.999 - 0.0434i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.999 - 0.0434i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(67\)
\( \varepsilon \)  =  $-0.999 - 0.0434i$
motivic weight  =  \(0\)
character  :  $\chi_{67} (28, \cdot )$
Sato-Tate  :  $\mu(66)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 67,\ (1:\ ),\ -0.999 - 0.0434i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.008589387832 - 0.3953721114i$
$L(\frac12,\chi)$  $\approx$  $0.008589387832 - 0.3953721114i$
$L(\chi,1)$  $\approx$  0.5280757953 - 0.2043146002i
$L(1,\chi)$  $\approx$  0.5280757953 - 0.2043146002i

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

−31.82495641601437387207380810652, −31.339779201525511415939931421646, −29.94890052821234487679746887906, −28.85899305036619594138965734527, −27.66222827544688299052545125240, −27.056624003900441636141052229569, −26.01433689373593953590678874747, −25.13432869880252061943212332021, −22.86626639945302746955319965233, −22.05187163998489109480395417941, −21.128403955290622484932762324270, −19.96002269854037676892290245900, −18.78588242375751356563932809227, −17.955792692327665879855184020097, −16.165010320135730596780156300348, −15.58107160435269417719623277189, −13.99895428044876949935462787291, −12.16665228190145362405410571758, −11.03279377715031164165080213928, −10.17654662701997549363612802420, −8.89210699644842974239869172259, −7.75491603217297205744204956266, −5.64864927143567490668322563110, −3.53399403279945784911792756726, −2.74834811206079555617116199100, 0.24871813605569732879751090719, 1.714612632193628394804375283983, 4.49133094454330243880727531760, 6.27675883425852483629690268088, 7.45421677485047489243275999919, 8.33676569117969285883072254066, 9.688889592873566042593374925202, 11.34190656996234189443406314509, 12.88397318024677249538883969038, 13.892564038256566413479820615789, 15.38902195576171075781827404761, 16.71625826008865261936547015264, 17.45117783105409976059612073139, 18.77379802623434361344951075442, 19.78424332822170820343621914030, 20.53291912340674890789428958158, 23.02092427886639919154160071736, 23.8253901193137730614238016046, 24.39584065526813088969702345899, 25.859800422968565540382612991172, 26.41686061703386958102522458443, 28.2253284751887438982535054838, 28.60262280182748080801735009104, 30.034121514340116524143531675725, 31.282864821212061693097406088854

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