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Properties

Label 2-201-3.2-c2-0-19
Degree $2$
Conductor $201$
Sign $0.989 + 0.143i$
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  − 1.74i·2-s + (2.96 + 0.431i)3-s + 0.964·4-s + 9.03i·5-s + (0.751 − 5.17i)6-s − 5.84·7-s − 8.64i·8-s + (8.62 + 2.56i)9-s + 15.7·10-s − 0.800i·11-s + (2.86 + 0.416i)12-s + 17.6·13-s + 10.1i·14-s + (−3.89 + 26.8i)15-s − 11.2·16-s + 16.0i·17-s + ⋯
L(s)  = 1  − 0.871i·2-s + (0.989 + 0.143i)3-s + 0.241·4-s + 1.80i·5-s + (0.125 − 0.862i)6-s − 0.834·7-s − 1.08i·8-s + (0.958 + 0.284i)9-s + 1.57·10-s − 0.0727i·11-s + (0.238 + 0.0346i)12-s + 1.35·13-s + 0.727i·14-s + (−0.259 + 1.78i)15-s − 0.700·16-s + 0.943i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.989 + 0.143i$
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: \((2,\ 201,\ (\ :1),\ 0.989 + 0.143i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.28651 - 0.165250i\)
\(L(\frac12)\) \(\approx\) \(2.28651 - 0.165250i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.96 - 0.431i)T \)
67 \( 1 - 8.18T \)
good2 \( 1 + 1.74iT - 4T^{2} \)
5 \( 1 - 9.03iT - 25T^{2} \)
7 \( 1 + 5.84T + 49T^{2} \)
11 \( 1 + 0.800iT - 121T^{2} \)
13 \( 1 - 17.6T + 169T^{2} \)
17 \( 1 - 16.0iT - 289T^{2} \)
19 \( 1 - 12.7T + 361T^{2} \)
23 \( 1 + 29.2iT - 529T^{2} \)
29 \( 1 - 4.51iT - 841T^{2} \)
31 \( 1 - 2.95T + 961T^{2} \)
37 \( 1 + 44.9T + 1.36e3T^{2} \)
41 \( 1 + 16.2iT - 1.68e3T^{2} \)
43 \( 1 + 66.8T + 1.84e3T^{2} \)
47 \( 1 + 52.7iT - 2.20e3T^{2} \)
53 \( 1 + 39.4iT - 2.80e3T^{2} \)
59 \( 1 + 11.8iT - 3.48e3T^{2} \)
61 \( 1 + 75.9T + 3.72e3T^{2} \)
71 \( 1 + 99.0iT - 5.04e3T^{2} \)
73 \( 1 - 142.T + 5.32e3T^{2} \)
79 \( 1 - 7.34T + 6.24e3T^{2} \)
83 \( 1 - 96.7iT - 6.88e3T^{2} \)
89 \( 1 - 46.6iT - 7.92e3T^{2} \)
97 \( 1 - 97.0T + 9.40e3T^{2} \)
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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

−12.11407770075932471509724135129, −10.83858586375997978212099258699, −10.48124238148575848617373845774, −9.614632148063107864700624196059, −8.255568494201638369865428393970, −6.89573799374787671302724511593, −6.40096284493358881413657524703, −3.59443110500285748676880439799, −3.29927572738570600664225344749, −2.03623773032457835325784755387, 1.44457957429253788899544742185, 3.37709537374777762306914977334, 4.90500147509465333243487799531, 6.05205942180877980716564316855, 7.30594915585734100770195408899, 8.244428113751616021338285208926, 8.985669884236450657890041405860, 9.743900184166166121959148453980, 11.51790719528372178766126863134, 12.50251067281596203989745262647

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