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Properties

Label 2-201-3.2-c2-0-41
Degree $2$
Conductor $201$
Sign $-0.711 - 0.703i$
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  − 2.26i·2-s + (−2.13 − 2.10i)3-s − 1.14·4-s − 6.81i·5-s + (−4.78 + 4.83i)6-s − 3.54·7-s − 6.48i·8-s + (0.101 + 8.99i)9-s − 15.4·10-s − 2.44i·11-s + (2.43 + 2.40i)12-s + 10.2·13-s + 8.03i·14-s + (−14.3 + 14.5i)15-s − 19.2·16-s − 4.43i·17-s + ⋯
L(s)  = 1  − 1.13i·2-s + (−0.711 − 0.703i)3-s − 0.285·4-s − 1.36i·5-s + (−0.797 + 0.806i)6-s − 0.506·7-s − 0.810i·8-s + (0.0112 + 0.999i)9-s − 1.54·10-s − 0.222i·11-s + (0.202 + 0.200i)12-s + 0.789·13-s + 0.574i·14-s + (−0.957 + 0.968i)15-s − 1.20·16-s − 0.261i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.703i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.711 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.711 - 0.703i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.711 - 0.703i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.388821 + 0.946224i\)
\(L(\frac12)\) \(\approx\) \(0.388821 + 0.946224i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.13 + 2.10i)T \)
67 \( 1 - 8.18T \)
good2 \( 1 + 2.26iT - 4T^{2} \)
5 \( 1 + 6.81iT - 25T^{2} \)
7 \( 1 + 3.54T + 49T^{2} \)
11 \( 1 + 2.44iT - 121T^{2} \)
13 \( 1 - 10.2T + 169T^{2} \)
17 \( 1 + 4.43iT - 289T^{2} \)
19 \( 1 + 0.511T + 361T^{2} \)
23 \( 1 - 36.8iT - 529T^{2} \)
29 \( 1 - 2.56iT - 841T^{2} \)
31 \( 1 + 20.9T + 961T^{2} \)
37 \( 1 - 40.3T + 1.36e3T^{2} \)
41 \( 1 + 80.7iT - 1.68e3T^{2} \)
43 \( 1 - 6.29T + 1.84e3T^{2} \)
47 \( 1 - 1.45iT - 2.20e3T^{2} \)
53 \( 1 + 60.2iT - 2.80e3T^{2} \)
59 \( 1 + 99.5iT - 3.48e3T^{2} \)
61 \( 1 + 30.2T + 3.72e3T^{2} \)
71 \( 1 + 43.8iT - 5.04e3T^{2} \)
73 \( 1 - 8.71T + 5.32e3T^{2} \)
79 \( 1 + 41.8T + 6.24e3T^{2} \)
83 \( 1 - 16.0iT - 6.88e3T^{2} \)
89 \( 1 - 70.2iT - 7.92e3T^{2} \)
97 \( 1 - 95.3T + 9.40e3T^{2} \)
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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

−11.64886690996419833818571692071, −11.00966234242594665164075176453, −9.790395725174127929652125649591, −8.853445903242352177090653625113, −7.53314880353390800819822891377, −6.23689395071069799328982614841, −5.11974608120387638106947406228, −3.64144689133126256202314716770, −1.79905997975692455226967419976, −0.64182582869130281910350193959, 2.94956068838068378554130516433, 4.46128646539144476755633453875, 6.05324288427424898170828142672, 6.37953793338853532433783597718, 7.40183205056412950252979764635, 8.747828581199417547455980931839, 10.03574673548569098640173199958, 10.85685941595419008982329255875, 11.52539011014284274740490489794, 12.89618265321577217500848492517

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