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Properties

Label 2-201-67.29-c1-0-0
Degree $2$
Conductor $201$
Sign $-0.912 - 0.408i$
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  + (−0.947 + 1.64i)2-s − 3-s + (−0.793 − 1.37i)4-s + 1.30·5-s + (0.947 − 1.64i)6-s + (2.53 + 4.39i)7-s − 0.781·8-s + 9-s + (−1.23 + 2.14i)10-s + (−1.88 − 3.26i)11-s + (0.793 + 1.37i)12-s + (−2.77 + 4.80i)13-s − 9.61·14-s − 1.30·15-s + (2.32 − 4.03i)16-s + (−2.28 + 3.95i)17-s + ⋯
L(s)  = 1  + (−0.669 + 1.15i)2-s − 0.577·3-s + (−0.396 − 0.687i)4-s + 0.584·5-s + (0.386 − 0.669i)6-s + (0.958 + 1.66i)7-s − 0.276·8-s + 0.333·9-s + (−0.391 + 0.677i)10-s + (−0.568 − 0.983i)11-s + (0.229 + 0.396i)12-s + (−0.769 + 1.33i)13-s − 2.56·14-s − 0.337·15-s + (0.581 − 1.00i)16-s + (−0.554 + 0.959i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.912 - 0.408i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ -0.912 - 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151459 + 0.709237i\)
\(L(\frac12)\) \(\approx\) \(0.151459 + 0.709237i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
67 \( 1 + (-8.01 + 1.67i)T \)
good2 \( 1 + (0.947 - 1.64i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 - 1.30T + 5T^{2} \)
7 \( 1 + (-2.53 - 4.39i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.88 + 3.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.77 - 4.80i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.28 - 3.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.737 + 1.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.39 + 2.41i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.62 - 4.55i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.883 - 1.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.48 + 6.04i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.25 - 7.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.89T + 43T^{2} \)
47 \( 1 + (2.59 + 4.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.37T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + (-6.21 + 10.7i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (4.98 + 8.63i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.00 + 3.47i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.880 + 1.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.08 + 1.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 7.24T + 89T^{2} \)
97 \( 1 + (2.81 - 4.88i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.74939868262364007071774486791, −11.81168285632363210904973904162, −10.97416324023771764972654697974, −9.470326778275161143592988110634, −8.789273474605939011347821438027, −7.945808529366539647599889595811, −6.55561172673298462952429958114, −5.81204261836985796290482038384, −4.94879545280088378781312854877, −2.29576934943475582805826679881, 0.856957591903110563850076569670, 2.41167964359029363316903996331, 4.29330134736167621842374783809, 5.46767236202195427894037367555, 7.18178226595374186454864876969, 7.984316512559337886253212558615, 9.779659676922449645601380943291, 10.07980371401741363942203814014, 10.92455338253596398266293122997, 11.69309609613275388958537625314

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