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Properties

Label 1-67-67.12-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: $\chi_{67} (12, \cdot )$
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

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

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