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Properties

Label 2-201-3.2-c2-0-34
Degree $2$
Conductor $201$
Sign $-0.256 + 0.966i$
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  − 1.01i·2-s + (−0.770 + 2.89i)3-s + 2.97·4-s − 8.93i·5-s + (2.93 + 0.778i)6-s − 6.17·7-s − 7.05i·8-s + (−7.81 − 4.46i)9-s − 9.02·10-s − 2.41i·11-s + (−2.29 + 8.63i)12-s − 5.04·13-s + 6.24i·14-s + (25.9 + 6.87i)15-s + 4.78·16-s − 12.1i·17-s + ⋯
L(s)  = 1  − 0.505i·2-s + (−0.256 + 0.966i)3-s + 0.744·4-s − 1.78i·5-s + (0.488 + 0.129i)6-s − 0.882·7-s − 0.881i·8-s + (−0.868 − 0.496i)9-s − 0.902·10-s − 0.219i·11-s + (−0.191 + 0.719i)12-s − 0.388·13-s + 0.446i·14-s + (1.72 + 0.458i)15-s + 0.298·16-s − 0.717i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.256 + 0.966i$
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: \((2,\ 201,\ (\ :1),\ -0.256 + 0.966i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.794680 - 1.03331i\)
\(L(\frac12)\) \(\approx\) \(0.794680 - 1.03331i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.770 - 2.89i)T \)
67 \( 1 - 8.18T \)
good2 \( 1 + 1.01iT - 4T^{2} \)
5 \( 1 + 8.93iT - 25T^{2} \)
7 \( 1 + 6.17T + 49T^{2} \)
11 \( 1 + 2.41iT - 121T^{2} \)
13 \( 1 + 5.04T + 169T^{2} \)
17 \( 1 + 12.1iT - 289T^{2} \)
19 \( 1 - 10.7T + 361T^{2} \)
23 \( 1 + 28.0iT - 529T^{2} \)
29 \( 1 - 26.8iT - 841T^{2} \)
31 \( 1 - 21.7T + 961T^{2} \)
37 \( 1 - 42.5T + 1.36e3T^{2} \)
41 \( 1 - 69.7iT - 1.68e3T^{2} \)
43 \( 1 - 28.5T + 1.84e3T^{2} \)
47 \( 1 + 64.0iT - 2.20e3T^{2} \)
53 \( 1 - 4.09iT - 2.80e3T^{2} \)
59 \( 1 - 39.4iT - 3.48e3T^{2} \)
61 \( 1 + 48.4T + 3.72e3T^{2} \)
71 \( 1 + 17.9iT - 5.04e3T^{2} \)
73 \( 1 - 105.T + 5.32e3T^{2} \)
79 \( 1 - 57.9T + 6.24e3T^{2} \)
83 \( 1 - 104. iT - 6.88e3T^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 + 159.T + 9.40e3T^{2} \)
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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

−12.04020242529883803121126576085, −11.00764441078963169342357908416, −9.812777848591112055299524597394, −9.367443098700660545066279173500, −8.185707195690777190852781455059, −6.52913743029578339238627149380, −5.34919579243649805129341213886, −4.29567980208711245253387077295, −2.90889510546112565332606991451, −0.74037068627780078621670092367, 2.24450181107781583745926372550, 3.23245209363280854804373255786, 5.85943001152087192076481134499, 6.40915585961852754477420620925, 7.27987049222770231552363927952, 7.81977845352577168478969320397, 9.730950307212783831610272659845, 10.77910369580657446386906333420, 11.48569746728920875623468050950, 12.39070102076048874397450112275

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