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Properties

Label 2-201-201.164-c1-0-3
Degree $2$
Conductor $201$
Sign $0.612 - 0.790i$
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  + (0.725 + 1.25i)2-s + (−1.60 − 0.650i)3-s + (−0.0538 + 0.0933i)4-s − 0.670·5-s + (−0.347 − 2.49i)6-s + (4.01 + 2.32i)7-s + 2.74·8-s + (2.15 + 2.08i)9-s + (−0.486 − 0.843i)10-s + (−0.0803 + 0.139i)11-s + (0.147 − 0.114i)12-s + (−3.16 + 1.82i)13-s + 6.73i·14-s + (1.07 + 0.436i)15-s + (2.10 + 3.64i)16-s + (5.25 − 3.03i)17-s + ⋯
L(s)  = 1  + (0.513 + 0.889i)2-s + (−0.926 − 0.375i)3-s + (−0.0269 + 0.0466i)4-s − 0.300·5-s + (−0.141 − 1.01i)6-s + (1.51 + 0.877i)7-s + 0.971·8-s + (0.718 + 0.695i)9-s + (−0.153 − 0.266i)10-s + (−0.0242 + 0.0419i)11-s + (0.0424 − 0.0331i)12-s + (−0.876 + 0.506i)13-s + 1.80i·14-s + (0.278 + 0.112i)15-s + (0.525 + 0.910i)16-s + (1.27 − 0.735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.612 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23257 + 0.604334i\)
\(L(\frac12)\) \(\approx\) \(1.23257 + 0.604334i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 + 0.650i)T \)
67 \( 1 + (-7.34 + 3.60i)T \)
good2 \( 1 + (-0.725 - 1.25i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.670T + 5T^{2} \)
7 \( 1 + (-4.01 - 2.32i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.0803 - 0.139i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.16 - 1.82i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.25 + 3.03i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.97 + 3.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.184 - 0.106i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.260 - 0.150i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.10 + 2.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.00 - 3.47i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.85 - 3.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 0.0314iT - 43T^{2} \)
47 \( 1 + (2.71 + 1.57i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 + 13.1iT - 59T^{2} \)
61 \( 1 + (5.55 - 3.20i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (-1.51 - 0.875i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.32 + 12.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.51 + 2.03i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.51 + 3.18i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + (-7.24 + 4.18i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.47640070550365129057898254155, −11.63722158689174614872022884388, −11.05098852842875622393442613255, −9.690805268651852681111678667406, −8.005006330473824105911942881736, −7.43538712207828024124957904422, −6.23148438029841852953629990164, −5.15835663312496025176536371812, −4.70056343041724780880115292340, −1.85479839803222441661595385699, 1.53419452208839638894676661395, 3.69453084749726759234414023215, 4.54205317338972408182358676158, 5.55810349816781725548241369777, 7.38274852302916693811443547204, 8.005073048231970401924836620384, 10.03986690509993643648040039248, 10.62188844027578978406279764638, 11.38657822840516905598841819183, 12.14763322904702269173488221310

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