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Properties

Label 2-201-67.29-c3-0-20
Degree $2$
Conductor $201$
Sign $-0.0856 + 0.996i$
Analytic cond. $11.8593$
Root an. cond. $3.44374$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.561 + 0.973i)2-s − 3·3-s + (3.36 + 5.83i)4-s − 9.21·5-s + (1.68 − 2.91i)6-s + (2.79 + 4.83i)7-s − 16.5·8-s + 9·9-s + (5.17 − 8.97i)10-s + (−15.6 − 27.0i)11-s + (−10.1 − 17.5i)12-s + (−35.5 + 61.5i)13-s − 6.27·14-s + 27.6·15-s + (−17.6 + 30.5i)16-s + (38.0 − 65.8i)17-s + ⋯
L(s)  = 1  + (−0.198 + 0.344i)2-s − 0.577·3-s + (0.421 + 0.729i)4-s − 0.824·5-s + (0.114 − 0.198i)6-s + (0.150 + 0.260i)7-s − 0.731·8-s + 0.333·9-s + (0.163 − 0.283i)10-s + (−0.428 − 0.741i)11-s + (−0.243 − 0.421i)12-s + (−0.758 + 1.31i)13-s − 0.119·14-s + 0.475·15-s + (−0.275 + 0.477i)16-s + (0.542 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0856 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0856 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.0856 + 0.996i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.0856 + 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.179104 - 0.195157i\)
\(L(\frac12)\) \(\approx\) \(0.179104 - 0.195157i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
67 \( 1 + (-282. - 470. i)T \)
good2 \( 1 + (0.561 - 0.973i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 9.21T + 125T^{2} \)
7 \( 1 + (-2.79 - 4.83i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (15.6 + 27.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (35.5 - 61.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-38.0 + 65.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-44.2 + 76.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-40.9 + 70.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (4.66 + 8.08i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (109. + 188. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-103. + 179. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-7.08 - 12.2i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 257.T + 7.95e4T^{2} \)
47 \( 1 + (115. + 199. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 147.T + 1.48e5T^{2} \)
59 \( 1 - 211.T + 2.05e5T^{2} \)
61 \( 1 + (286. - 495. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (-27.4 - 47.5i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (202. - 350. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (162. + 281. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (653. - 1.13e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.64e3T + 7.04e5T^{2} \)
97 \( 1 + (684. - 1.18e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.51586024925361715661499427288, −11.33964180938938532931661930890, −9.623052628027708210935781110296, −8.566350418716628799946821186928, −7.48810172173927708649477145464, −6.85703525739489264528820330225, −5.43571274321593625407779714607, −4.09760581762792364773201260780, −2.63839444570367891984990286519, −0.13223146808090782013563242821, 1.43802738975883395017531410839, 3.28372845044655881906793386500, 4.92723041154703978296582789851, 5.86064260114761071826904226645, 7.26287842475978517237001218866, 8.019548964761293884657465010844, 9.772892583945704523217000023044, 10.34117225445237412736696188107, 11.20164175720500808192945063037, 12.16053651483802222389853736314

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