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Properties

Label 2-201-3.2-c2-0-31
Degree $2$
Conductor $201$
Sign $-0.949 + 0.313i$
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  − 1.16i·2-s + (−2.84 + 0.941i)3-s + 2.64·4-s + 5.51i·5-s + (1.09 + 3.31i)6-s − 12.7·7-s − 7.72i·8-s + (7.22 − 5.36i)9-s + 6.41·10-s − 16.0i·11-s + (−7.54 + 2.49i)12-s − 10.1·13-s + 14.8i·14-s + (−5.19 − 15.7i)15-s + 1.61·16-s − 26.6i·17-s + ⋯
L(s)  = 1  − 0.581i·2-s + (−0.949 + 0.313i)3-s + 0.662·4-s + 1.10i·5-s + (0.182 + 0.551i)6-s − 1.81·7-s − 0.966i·8-s + (0.803 − 0.595i)9-s + 0.641·10-s − 1.45i·11-s + (−0.628 + 0.207i)12-s − 0.779·13-s + 1.05i·14-s + (−0.346 − 1.04i)15-s + 0.100·16-s − 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.949 + 0.313i$
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: \((2,\ 201,\ (\ :1),\ -0.949 + 0.313i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0556098 - 0.345457i\)
\(L(\frac12)\) \(\approx\) \(0.0556098 - 0.345457i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.84 - 0.941i)T \)
67 \( 1 - 8.18T \)
good2 \( 1 + 1.16iT - 4T^{2} \)
5 \( 1 - 5.51iT - 25T^{2} \)
7 \( 1 + 12.7T + 49T^{2} \)
11 \( 1 + 16.0iT - 121T^{2} \)
13 \( 1 + 10.1T + 169T^{2} \)
17 \( 1 + 26.6iT - 289T^{2} \)
19 \( 1 + 22.2T + 361T^{2} \)
23 \( 1 - 19.4iT - 529T^{2} \)
29 \( 1 - 11.3iT - 841T^{2} \)
31 \( 1 + 27.2T + 961T^{2} \)
37 \( 1 + 21.4T + 1.36e3T^{2} \)
41 \( 1 + 4.18iT - 1.68e3T^{2} \)
43 \( 1 + 27.0T + 1.84e3T^{2} \)
47 \( 1 - 60.2iT - 2.20e3T^{2} \)
53 \( 1 + 64.4iT - 2.80e3T^{2} \)
59 \( 1 + 5.34iT - 3.48e3T^{2} \)
61 \( 1 - 72.1T + 3.72e3T^{2} \)
71 \( 1 - 2.89iT - 5.04e3T^{2} \)
73 \( 1 + 22.4T + 5.32e3T^{2} \)
79 \( 1 + 101.T + 6.24e3T^{2} \)
83 \( 1 + 75.5iT - 6.88e3T^{2} \)
89 \( 1 - 106. iT - 7.92e3T^{2} \)
97 \( 1 + 130.T + 9.40e3T^{2} \)
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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

−11.54795989584286778571758699901, −10.89335414239386671204060957868, −10.10980291774895839371848194263, −9.365189137896206598275792718134, −7.08178406247524182166667551220, −6.65972187712183590342711705227, −5.68527577308451551748619762106, −3.57468353765229015452051945397, −2.83962306213895578962115729568, −0.20211035615214667221888906341, 2.04285216212478209623509028389, 4.32192175922503689229396395392, 5.58542079549344360213025358116, 6.53676006786180452443890894345, 7.15887868923956730497052193601, 8.530533444001858673271326798659, 9.887704367929990009019545198683, 10.56060296755726055517354185862, 12.14726387284916637273346227614, 12.58934371968245495821926895979

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