Properties

Degree 1
Conductor 67
Sign $0.764 + 0.644i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.888 + 0.458i)2-s + (0.959 − 0.281i)3-s + (0.580 + 0.814i)4-s + (0.654 + 0.755i)5-s + (0.981 + 0.189i)6-s + (−0.0475 − 0.998i)7-s + (0.142 + 0.989i)8-s + (0.841 − 0.540i)9-s + (0.235 + 0.971i)10-s + (−0.981 + 0.189i)11-s + (0.786 + 0.618i)12-s + (−0.928 + 0.371i)13-s + (0.415 − 0.909i)14-s + (0.841 + 0.540i)15-s + (−0.327 + 0.945i)16-s + (0.580 − 0.814i)17-s + ⋯
L(s,χ)  = 1  + (0.888 + 0.458i)2-s + (0.959 − 0.281i)3-s + (0.580 + 0.814i)4-s + (0.654 + 0.755i)5-s + (0.981 + 0.189i)6-s + (−0.0475 − 0.998i)7-s + (0.142 + 0.989i)8-s + (0.841 − 0.540i)9-s + (0.235 + 0.971i)10-s + (−0.981 + 0.189i)11-s + (0.786 + 0.618i)12-s + (−0.928 + 0.371i)13-s + (0.415 − 0.909i)14-s + (0.841 + 0.540i)15-s + (−0.327 + 0.945i)16-s + (0.580 − 0.814i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.764 + 0.644i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.764 + 0.644i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(67\)
\( \varepsilon \)  =  $0.764 + 0.644i$
motivic weight  =  \(0\)
character  :  $\chi_{67} (32, \cdot )$
Sato-Tate  :  $\mu(66)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 67,\ (1:\ ),\ 0.764 + 0.644i)$
$L(\chi,\frac{1}{2})$  $\approx$  $3.484670726 + 1.272875736i$
$L(\frac12,\chi)$  $\approx$  $3.484670726 + 1.272875736i$
$L(\chi,1)$  $\approx$  2.323573412 + 0.6079043976i
$L(1,\chi)$  $\approx$  2.323573412 + 0.6079043976i

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

−31.757819348266409597508654634159, −30.86372508215018866704312138681, −29.497000362891559811543011359345, −28.587614295269637743863920556708, −27.46047534379612630720241207831, −25.75932465116312333350667238341, −24.859164188913421608560181705177, −24.13369011411145351193586352723, −22.40520086216068358858644959477, −21.240088083317464297705229112891, −20.88698422952637820996794935575, −19.54866982190856768998167919596, −18.513923779853512376269414058, −16.472478027300910019843038395761, −15.26123282197878400331062904519, −14.383857246691692334914797204678, −13.021940781640317647755205003356, −12.42621817669487973625385543674, −10.41675078521177322111983816872, −9.42491234906406482449415767715, −7.98994772417144045236751148825, −5.81113499053034100141861369460, −4.780362264100620952459983020068, −3.01660805750151208002766797495, −1.886350159323029111793014501242, 2.296703320421386128612101375301, 3.43086726105923905214602822305, 5.117495868063676994197919368914, 7.076751366606120844062343454846, 7.43056545615769854612224146641, 9.45481378112348116854022369460, 10.90345380955288664930090869879, 12.75796867522271663758463161768, 13.698848510070186996899455949618, 14.41589271829090595380834129274, 15.50902809414597210395677861036, 17.0261438084200927018381963417, 18.29057923525176640572608963507, 19.80383971986485659843788376305, 20.866712767893321654137602423743, 21.80458973862008142960781880260, 23.1671722693284655196263272260, 24.10986891467902125739804301876, 25.31988049088884272622301932870, 26.14500610129619593503566306652, 26.80404832008661559812897433751, 29.39362040503042105632989083719, 29.7623661610401896029811530253, 30.954272341522415857664275923218, 31.769477507195709964888399214711

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