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Properties

Label 2-201-201.38-c1-0-11
Degree $2$
Conductor $201$
Sign $0.110 - 0.993i$
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 + (−1.98 + 1.54i)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 + (−0.0258 − 0.184i)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 + (−0.809 + 0.631i)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 + (−0.00744 − 0.0533i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.110 - 0.993i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.110 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03245 + 0.924076i\)
\(L(\frac12)\) \(\approx\) \(1.03245 + 0.924076i\)
\(L(\frac{3}{2})\) 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

−12.91280295330303807360786224216, −11.54288950124974457048481637617, −10.48631867166246261842313780347, −9.444845060035930009915223386987, −8.441604514718597088150546265586, −7.72803308809125125419389007506, −7.02104900505653899868539645382, −5.26618053583072015820412625149, −4.05761787740296316183165884336, −2.25528810675453134332701811705, 1.86047836704722614063740648500, 2.42704001816360112970038528917, 4.41075368240795838974078389748, 5.97886558489783559622352874926, 7.41024381410431424835245543645, 8.726990330655468699130154811192, 9.015311367879091174135918959007, 10.24953332325065029332896587858, 11.33217059084451621621578770806, 11.98454329115469505859334745388

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