Properties

Label 1-67-67.44-r1-0-0
Degree $1$
Conductor $67$
Sign $0.764 - 0.644i$
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  + (0.888 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ 0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.484670726 - 1.272875736i\)
\(L(\frac12)\) \(\approx\) \(3.484670726 - 1.272875736i\)
\(L(1)\) \(\approx\) \(2.323573412 - 0.6079043976i\)
\(L(1)\) \(\approx\) \(2.323573412 - 0.6079043976i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.888 - 0.458i)T \)
3 \( 1 + (0.959 + 0.281i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (-0.0475 + 0.998i)T \)
11 \( 1 + (-0.981 - 0.189i)T \)
13 \( 1 + (-0.928 - 0.371i)T \)
17 \( 1 + (0.580 + 0.814i)T \)
19 \( 1 + (0.0475 + 0.998i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.928 + 0.371i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.995 - 0.0950i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.235 + 0.971i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (-0.981 + 0.189i)T \)
71 \( 1 + (0.580 - 0.814i)T \)
73 \( 1 + (0.981 - 0.189i)T \)
79 \( 1 + (0.786 - 0.618i)T \)
83 \( 1 + (-0.327 - 0.945i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
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

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

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