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Properties

Label 1-67-67.48-r1-0-0
Degree $1$
Conductor $67$
Sign $-0.930 + 0.367i$
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  + (−0.580 − 0.814i)2-s + (−0.841 + 0.540i)3-s + (−0.327 + 0.945i)4-s + (0.142 − 0.989i)5-s + (0.928 + 0.371i)6-s + (0.995 − 0.0950i)7-s + (0.959 − 0.281i)8-s + (0.415 − 0.909i)9-s + (−0.888 + 0.458i)10-s + (−0.928 + 0.371i)11-s + (−0.235 − 0.971i)12-s + (−0.723 + 0.690i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (−0.786 − 0.618i)16-s + (−0.327 − 0.945i)17-s + ⋯
L(s)  = 1  + (−0.580 − 0.814i)2-s + (−0.841 + 0.540i)3-s + (−0.327 + 0.945i)4-s + (0.142 − 0.989i)5-s + (0.928 + 0.371i)6-s + (0.995 − 0.0950i)7-s + (0.959 − 0.281i)8-s + (0.415 − 0.909i)9-s + (−0.888 + 0.458i)10-s + (−0.928 + 0.371i)11-s + (−0.235 − 0.971i)12-s + (−0.723 + 0.690i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (−0.786 − 0.618i)16-s + (−0.327 − 0.945i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.930 + 0.367i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ -0.930 + 0.367i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03508324830 - 0.1844318103i\)
\(L(\frac12)\) \(\approx\) \(-0.03508324830 - 0.1844318103i\)
\(L(1)\) \(\approx\) \(0.4574496845 - 0.1881065386i\)
\(L(1)\) \(\approx\) \(0.4574496845 - 0.1881065386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (-0.580 - 0.814i)T \)
3 \( 1 + (-0.841 + 0.540i)T \)
5 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (0.995 - 0.0950i)T \)
11 \( 1 + (-0.928 + 0.371i)T \)
13 \( 1 + (-0.723 + 0.690i)T \)
17 \( 1 + (-0.327 - 0.945i)T \)
19 \( 1 + (-0.995 - 0.0950i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.723 - 0.690i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.981 - 0.189i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.888 - 0.458i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (-0.928 - 0.371i)T \)
71 \( 1 + (-0.327 + 0.945i)T \)
73 \( 1 + (0.928 + 0.371i)T \)
79 \( 1 + (-0.235 - 0.971i)T \)
83 \( 1 + (-0.786 - 0.618i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
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

−32.798373204745565243882232822563, −31.17939853488306143133214948586, −30.00703445476972909575026180856, −28.9848643444202088794072951191, −27.79345511426015359851242877081, −26.92591883960652416700603199313, −25.76413630203711485763155385785, −24.50364735765182307629684379830, −23.73661714260080858648842936515, −22.699775386595246981040175513830, −21.53117459344644461018079719654, −19.49559201744902887402014194191, −18.36317512513938765728365070618, −17.82221688942442998430139786174, −16.76708657518829289870713758547, −15.26683971321282808089553034328, −14.33793953493801619582490792486, −12.82145722969367291966047703537, −10.89040796600703464613357679952, −10.50481846512085948763472531592, −8.29256474805224177017436264660, −7.308558739888929223367747760715, −6.09225277516197262929699091547, −4.972843473388416090400504684134, −1.98808514169445213110057096000, 0.125255842818208860748392843173, 1.89506089237735607784263599528, 4.3197726288881559714179198679, 5.14445619150975139104480257546, 7.49449410415835768078234210207, 8.96886750457466002656298601049, 10.03266164058112212186305424507, 11.31107937589871077573095455788, 12.14476382621818174994393811621, 13.42829649395917205132830287101, 15.44687802390167705258647932524, 16.82992744787899176317289793053, 17.40847447273629527131039100981, 18.55418138706221207460195999415, 20.25308445862782048465353776699, 21.02812390050763629966944485334, 21.77083802352205727845387997917, 23.29198939747761090832074942604, 24.36828874846142184582422041182, 26.00838665854712295272646285501, 27.2750113972914129485576235214, 27.83037282444425756950023010243, 28.883624468357545398257601815254, 29.54703898805800676999220694242, 31.148391264119147249056952928759

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