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Properties

Label 2-201-201.80-c1-0-3
Degree $2$
Conductor $201$
Sign $0.0674 - 0.997i$
Analytic cond. $1.60499$
Root an. cond. $1.26688$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.30 − 1.13i)3-s + (−0.471 + 1.94i)4-s + (−0.816 + 2.03i)7-s + (0.426 + 2.96i)9-s + (2.82 − 2.00i)12-s + (0.682 + 7.14i)13-s + (−3.55 − 1.83i)16-s + (2.43 − 0.974i)19-s + (3.38 − 1.74i)21-s + (−2.07 + 4.54i)25-s + (2.80 − 4.37i)27-s + (−3.57 − 2.54i)28-s + (0.332 − 3.48i)31-s + (−5.97 − 0.570i)36-s + (−5.22 − 9.05i)37-s + ⋯
L(s)  = 1  + (−0.755 − 0.654i)3-s + (−0.235 + 0.971i)4-s + (−0.308 + 0.770i)7-s + (0.142 + 0.989i)9-s + (0.814 − 0.580i)12-s + (0.189 + 1.98i)13-s + (−0.888 − 0.458i)16-s + (0.558 − 0.223i)19-s + (0.738 − 0.380i)21-s + (−0.415 + 0.909i)25-s + (0.540 − 0.841i)27-s + (−0.676 − 0.481i)28-s + (0.0597 − 0.625i)31-s + (−0.995 − 0.0950i)36-s + (−0.859 − 1.48i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0674 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0674 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.0674 - 0.997i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.0674 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.539775 + 0.504512i\)
\(L(\frac12)\) \(\approx\) \(0.539775 + 0.504512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.30 + 1.13i)T \)
67 \( 1 + (-6.05 - 5.51i)T \)
good2 \( 1 + (0.471 - 1.94i)T^{2} \)
5 \( 1 + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (0.816 - 2.03i)T + (-5.06 - 4.83i)T^{2} \)
11 \( 1 + (-10.9 + 1.04i)T^{2} \)
13 \( 1 + (-0.682 - 7.14i)T + (-12.7 + 2.46i)T^{2} \)
17 \( 1 + (15.1 - 7.78i)T^{2} \)
19 \( 1 + (-2.43 + 0.974i)T + (13.7 - 13.1i)T^{2} \)
23 \( 1 + (-21.3 - 8.54i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.332 + 3.48i)T + (-30.4 - 5.86i)T^{2} \)
37 \( 1 + (5.22 + 9.05i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.95 - 40.9i)T^{2} \)
43 \( 1 + (0.137 - 0.468i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + (36.9 - 29.0i)T^{2} \)
53 \( 1 + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (38.6 - 44.5i)T^{2} \)
61 \( 1 + (-15.4 - 0.737i)T + (60.7 + 5.79i)T^{2} \)
71 \( 1 + (63.1 + 32.5i)T^{2} \)
73 \( 1 + (0.725 - 15.2i)T + (-72.6 - 6.93i)T^{2} \)
79 \( 1 + (-14.3 + 10.2i)T + (25.8 - 74.6i)T^{2} \)
83 \( 1 + (-48.1 - 67.6i)T^{2} \)
89 \( 1 + (12.6 - 88.0i)T^{2} \)
97 \( 1 + (9.48 - 5.47i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.55100855146848121816979254807, −11.75996733933861935004222474546, −11.25213410170799582719264421685, −9.519006031192036580638426269482, −8.702616880089293348634005899004, −7.43716963682552027252896480863, −6.64873309335644835262498626826, −5.37662276055833470750244855674, −3.99585275977670568270311617056, −2.21071803507130972697458664743, 0.72377367653169585993288460551, 3.50452656730089193848518647675, 4.89918602832544584819965992199, 5.73650696496293715961495292886, 6.77865702005430971358970750171, 8.298213840208004222201245655403, 9.747399980141132007857325070668, 10.27474346761138180948761275390, 10.88916513680349583627080300810, 12.12237593670056808791142237119

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