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Properties

Label 1-67-67.28-r1-0-0
Degree $1$
Conductor $67$
Sign $-0.999 - 0.0434i$
Analytic cond. $7.20014$
Root an. cond. $7.20014$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.999 - 0.0434i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ -0.999 - 0.0434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008589387832 - 0.3953721114i\)
\(L(\frac12)\) \(\approx\) \(0.008589387832 - 0.3953721114i\)
\(L(1)\) \(\approx\) \(0.5280757953 - 0.2043146002i\)
\(L(1)\) \(\approx\) \(0.5280757953 - 0.2043146002i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.723 + 0.690i)T \)
3 \( 1 + (0.142 - 0.989i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.235 - 0.971i)T \)
11 \( 1 + (-0.580 + 0.814i)T \)
13 \( 1 + (0.327 + 0.945i)T \)
17 \( 1 + (0.0475 + 0.998i)T \)
19 \( 1 + (0.235 - 0.971i)T \)
23 \( 1 + (-0.786 - 0.618i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.327 - 0.945i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.888 + 0.458i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (0.928 - 0.371i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (-0.654 - 0.755i)T \)
61 \( 1 + (-0.580 - 0.814i)T \)
71 \( 1 + (0.0475 - 0.998i)T \)
73 \( 1 + (0.580 + 0.814i)T \)
79 \( 1 + (-0.981 - 0.189i)T \)
83 \( 1 + (-0.995 - 0.0950i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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