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Properties

Label 1-67-67.38-r1-0-0
Degree $1$
Conductor $67$
Sign $0.667 - 0.744i$
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  + (0.5 + 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s − 5-s + (−0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + 9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + 14-s + 15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s − 5-s + (−0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + 9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + 14-s + 15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.667 - 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.667 - 0.744i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ 0.667 - 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6810074740 - 0.3042317139i\)
\(L(\frac12)\) \(\approx\) \(0.6810074740 - 0.3042317139i\)
\(L(1)\) \(\approx\) \(0.7512384426 + 0.1577051546i\)
\(L(1)\) \(\approx\) \(0.7512384426 + 0.1577051546i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
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

−31.662558628188287243814363064692, −30.63261012936760442896219768931, −29.9040617185195413554333523040, −28.26449387887699286463649655299, −28.02202420453986381632179219950, −27.06274773957034568364486596669, −24.87982664334745961162416664224, −23.699553665515740370827658520652, −22.915753753597025997088078848539, −22.02967255902568043420152913862, −20.86571821762452884026011684249, −19.58855323712634491007562685155, −18.4774537723603339363446693944, −17.47607701129753836484755264747, −15.60407839487336089436821525913, −14.91244912601192965065026604534, −12.8889593667659695404067171531, −12.03425897527400506545833060125, −11.26579044910652061313989646192, −10.04289887145436619426864368884, −8.26010296180386642419264116570, −6.280231976110708617601990382, −4.97269422433706412783887676094, −3.792774676011730624974735138935, −1.605614027477762016544712379686, 0.392968661753235068133902321519, 3.907272198088155500716303514175, 4.748665435728808990981637705632, 6.44717073040779575553691573397, 7.34511709719885249237753894828, 8.7949787005019413646377449853, 11.031271429152391638770032900873, 11.76930160029831433054341407695, 13.23678186311826353809791194267, 14.4695173104064545173131230924, 15.96573475594861692800576793957, 16.5342570130870075984275853669, 17.683118437721936056811774503001, 18.95046119853194521723311259902, 20.70029461313036638090653572190, 22.02418294487774455926403746212, 22.94064644799917988655178787706, 23.9616419024873364602834261529, 24.30451253904436334254751113892, 26.34818585044432131444399124324, 27.06022694194801346339222393052, 28.03937235430777536166809515478, 29.72670999446343606950205950150, 30.53357101132164763694907273244, 31.75406057260301806634098342605

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