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Properties

Label 2-201-3.2-c2-0-32
Degree $2$
Conductor $201$
Sign $0.957 + 0.287i$
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  − 0.198i·2-s + (2.87 + 0.861i)3-s + 3.96·4-s − 3.72i·5-s + (0.171 − 0.571i)6-s + 1.68·7-s − 1.58i·8-s + (7.51 + 4.94i)9-s − 0.739·10-s − 13.1i·11-s + (11.3 + 3.41i)12-s − 13.3·13-s − 0.334i·14-s + (3.20 − 10.6i)15-s + 15.5·16-s + 27.8i·17-s + ⋯
L(s)  = 1  − 0.0993i·2-s + (0.957 + 0.287i)3-s + 0.990·4-s − 0.744i·5-s + (0.0285 − 0.0952i)6-s + 0.240·7-s − 0.197i·8-s + (0.835 + 0.549i)9-s − 0.0739·10-s − 1.19i·11-s + (0.948 + 0.284i)12-s − 1.02·13-s − 0.0238i·14-s + (0.213 − 0.712i)15-s + 0.970·16-s + 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.957 + 0.287i$
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: \((2,\ 201,\ (\ :1),\ 0.957 + 0.287i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.47581 - 0.363000i\)
\(L(\frac12)\) \(\approx\) \(2.47581 - 0.363000i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.87 - 0.861i)T \)
67 \( 1 - 8.18T \)
good2 \( 1 + 0.198iT - 4T^{2} \)
5 \( 1 + 3.72iT - 25T^{2} \)
7 \( 1 - 1.68T + 49T^{2} \)
11 \( 1 + 13.1iT - 121T^{2} \)
13 \( 1 + 13.3T + 169T^{2} \)
17 \( 1 - 27.8iT - 289T^{2} \)
19 \( 1 + 23.2T + 361T^{2} \)
23 \( 1 + 15.7iT - 529T^{2} \)
29 \( 1 - 54.4iT - 841T^{2} \)
31 \( 1 + 14.4T + 961T^{2} \)
37 \( 1 - 20.9T + 1.36e3T^{2} \)
41 \( 1 + 76.2iT - 1.68e3T^{2} \)
43 \( 1 - 29.9T + 1.84e3T^{2} \)
47 \( 1 - 57.3iT - 2.20e3T^{2} \)
53 \( 1 - 20.1iT - 2.80e3T^{2} \)
59 \( 1 + 2.33iT - 3.48e3T^{2} \)
61 \( 1 + 120.T + 3.72e3T^{2} \)
71 \( 1 - 96.0iT - 5.04e3T^{2} \)
73 \( 1 + 96.7T + 5.32e3T^{2} \)
79 \( 1 + 83.5T + 6.24e3T^{2} \)
83 \( 1 + 138. iT - 6.88e3T^{2} \)
89 \( 1 - 1.86iT - 7.92e3T^{2} \)
97 \( 1 - 45.4T + 9.40e3T^{2} \)
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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

−12.53518618359878742840578168309, −10.90748357448162918023316261789, −10.44105611054382881571212318766, −8.933351283236728522102702742773, −8.350330164040025130321666225305, −7.23597091008056452767186250905, −5.90294315037266424191576663004, −4.41786799078302109605379148607, −3.05708123468127603615547831008, −1.69407915396011751170046434188, 2.08677116603482719265884472379, 2.87558894973846598825536624317, 4.60339617330755123744298815133, 6.45689715047835012350358314949, 7.29386157335815355174139530845, 7.82961342897009035931311847533, 9.448447059953281078643401614139, 10.15803028280078019410268735656, 11.40557984363511376944768541517, 12.21192865354872222934296250024

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