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Properties

Label 2-201-67.38-c2-0-22
Degree $2$
Conductor $201$
Sign $-0.999 + 0.0433i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.415 − 0.240i)2-s − 1.73i·3-s + (−1.88 − 3.26i)4-s − 7.03i·5-s + (−0.415 + 0.720i)6-s + (1.25 − 0.726i)7-s + 3.73i·8-s − 2.99·9-s + (−1.68 + 2.92i)10-s + (−2.57 + 1.48i)11-s + (−5.65 + 3.26i)12-s + (2.98 + 1.72i)13-s − 0.697·14-s − 12.1·15-s + (−6.64 + 11.5i)16-s + (−3.35 + 5.80i)17-s + ⋯
L(s)  = 1  + (−0.207 − 0.120i)2-s − 0.577i·3-s + (−0.471 − 0.816i)4-s − 1.40i·5-s + (−0.0692 + 0.120i)6-s + (0.179 − 0.103i)7-s + 0.466i·8-s − 0.333·9-s + (−0.168 + 0.292i)10-s + (−0.234 + 0.135i)11-s + (−0.471 + 0.272i)12-s + (0.229 + 0.132i)13-s − 0.0498·14-s − 0.812·15-s + (−0.415 + 0.719i)16-s + (−0.197 + 0.341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0433i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.999 + 0.0433i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.999 + 0.0433i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0200129 - 0.922117i\)
\(L(\frac12)\) \(\approx\) \(0.0200129 - 0.922117i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
67 \( 1 + (-51.7 - 42.4i)T \)
good2 \( 1 + (0.415 + 0.240i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + 7.03iT - 25T^{2} \)
7 \( 1 + (-1.25 + 0.726i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (2.57 - 1.48i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-2.98 - 1.72i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (3.35 - 5.80i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (3.81 - 6.61i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-14.3 + 24.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (23.1 + 40.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-5.39 + 3.11i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (10.8 - 18.7i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-16.2 + 9.35i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 3.54iT - 1.84e3T^{2} \)
47 \( 1 + (17.5 + 30.3i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 52.6iT - 2.80e3T^{2} \)
59 \( 1 - 32.9T + 3.48e3T^{2} \)
61 \( 1 + (-2.92 - 1.68i)T + (1.86e3 + 3.22e3i)T^{2} \)
71 \( 1 + (46.4 + 80.5i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-38.5 + 66.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-58.6 + 33.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-8.49 + 14.7i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 20.9T + 7.92e3T^{2} \)
97 \( 1 + (75.2 + 43.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.82669196380795693437502236127, −10.73238074152701154186619013280, −9.611748004770844890912987255469, −8.718854697479642750366289563740, −8.013899523171348512072324824477, −6.37266454076457765116418908693, −5.28994540669667564151787179694, −4.34096624930566834137878813661, −1.86718057253829270965748320130, −0.57658713627027205799038059246, 2.84849683590487059921302928715, 3.75224501465645426186338710510, 5.23361861738031517169377478779, 6.77029631637003633474579017986, 7.61410811594005427901449407247, 8.790246424958761366690934734510, 9.701968288861309867459884092743, 10.84044084890621996237969911237, 11.42619327534614164658585835271, 12.74769814242046142701826671261

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