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Properties

Label 2-201-67.66-c2-0-17
Degree $2$
Conductor $201$
Sign $0.990 + 0.139i$
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  + 3.21i·2-s + 1.73i·3-s − 6.34·4-s − 8.83i·5-s − 5.57·6-s − 10.0i·7-s − 7.55i·8-s − 2.99·9-s + 28.4·10-s − 5.71i·11-s − 10.9i·12-s − 8.59i·13-s + 32.1·14-s + 15.3·15-s − 1.07·16-s − 8.62·17-s + ⋯
L(s)  = 1  + 1.60i·2-s + 0.577i·3-s − 1.58·4-s − 1.76i·5-s − 0.928·6-s − 1.42i·7-s − 0.944i·8-s − 0.333·9-s + 2.84·10-s − 0.519i·11-s − 0.916i·12-s − 0.660i·13-s + 2.29·14-s + 1.02·15-s − 0.0674·16-s − 0.507·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.990 + 0.139i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.990 + 0.139i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.00982 - 0.0707403i\)
\(L(\frac12)\) \(\approx\) \(1.00982 - 0.0707403i\)
\(L(2)\) 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 \( 1 + (9.34 - 66.3i)T \)
good2 \( 1 - 3.21iT - 4T^{2} \)
5 \( 1 + 8.83iT - 25T^{2} \)
7 \( 1 + 10.0iT - 49T^{2} \)
11 \( 1 + 5.71iT - 121T^{2} \)
13 \( 1 + 8.59iT - 169T^{2} \)
17 \( 1 + 8.62T + 289T^{2} \)
19 \( 1 + 32.1T + 361T^{2} \)
23 \( 1 - 10.2T + 529T^{2} \)
29 \( 1 - 26.5T + 841T^{2} \)
31 \( 1 - 37.6iT - 961T^{2} \)
37 \( 1 - 33.0T + 1.36e3T^{2} \)
41 \( 1 - 5.22iT - 1.68e3T^{2} \)
43 \( 1 - 5.84iT - 1.84e3T^{2} \)
47 \( 1 - 79.8T + 2.20e3T^{2} \)
53 \( 1 + 54.0iT - 2.80e3T^{2} \)
59 \( 1 - 95.8T + 3.48e3T^{2} \)
61 \( 1 + 42.8iT - 3.72e3T^{2} \)
71 \( 1 - 62.2T + 5.04e3T^{2} \)
73 \( 1 + 39.4T + 5.32e3T^{2} \)
79 \( 1 + 109. iT - 6.24e3T^{2} \)
83 \( 1 + 121.T + 6.88e3T^{2} \)
89 \( 1 + 89.2T + 7.92e3T^{2} \)
97 \( 1 + 45.9iT - 9.40e3T^{2} \)
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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

−12.69334832899400603546779184144, −11.04026931476432696783636711719, −9.930542825652591299972975350681, −8.600011010924722271707900995554, −8.420526998670794596731602090298, −7.08298836329010043147814863206, −5.85520862675034037928818047770, −4.77198270424016903469932411773, −4.16241528933343323894374607894, −0.56890846365715482644835601632, 2.29201736167673064626769040889, 2.53935970432331439131740974793, 4.14902443861282048660101762832, 6.08534218909733370214362253632, 6.99678494973052039841737450319, 8.574586449890406590790024242942, 9.577758447761584649607535432410, 10.63273597173044290924932288404, 11.30969207702620825044421413608, 12.03689177017395853253678012660

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