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Properties

Label 2-201-3.2-c2-0-30
Degree $2$
Conductor $201$
Sign $0.800 - 0.598i$
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  + 3.85i·2-s + (2.40 − 1.79i)3-s − 10.8·4-s − 6.40i·5-s + (6.92 + 9.26i)6-s + 10.8·7-s − 26.4i·8-s + (2.54 − 8.63i)9-s + 24.6·10-s − 13.5i·11-s + (−26.1 + 19.5i)12-s − 6.76·13-s + 41.6i·14-s + (−11.5 − 15.3i)15-s + 58.6·16-s + 9.98i·17-s + ⋯
L(s)  = 1  + 1.92i·2-s + (0.800 − 0.598i)3-s − 2.71·4-s − 1.28i·5-s + (1.15 + 1.54i)6-s + 1.54·7-s − 3.30i·8-s + (0.283 − 0.959i)9-s + 2.46·10-s − 1.23i·11-s + (−2.17 + 1.62i)12-s − 0.520·13-s + 2.97i·14-s + (−0.766 − 1.02i)15-s + 3.66·16-s + 0.587i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.800 - 0.598i$
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: \((2,\ 201,\ (\ :1),\ 0.800 - 0.598i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.70726 + 0.567522i\)
\(L(\frac12)\) \(\approx\) \(1.70726 + 0.567522i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.40 + 1.79i)T \)
67 \( 1 - 8.18T \)
good2 \( 1 - 3.85iT - 4T^{2} \)
5 \( 1 + 6.40iT - 25T^{2} \)
7 \( 1 - 10.8T + 49T^{2} \)
11 \( 1 + 13.5iT - 121T^{2} \)
13 \( 1 + 6.76T + 169T^{2} \)
17 \( 1 - 9.98iT - 289T^{2} \)
19 \( 1 + 11.7T + 361T^{2} \)
23 \( 1 - 30.9iT - 529T^{2} \)
29 \( 1 + 6.49iT - 841T^{2} \)
31 \( 1 - 6.53T + 961T^{2} \)
37 \( 1 - 66.8T + 1.36e3T^{2} \)
41 \( 1 - 54.8iT - 1.68e3T^{2} \)
43 \( 1 + 18.7T + 1.84e3T^{2} \)
47 \( 1 + 16.4iT - 2.20e3T^{2} \)
53 \( 1 + 40.5iT - 2.80e3T^{2} \)
59 \( 1 - 33.3iT - 3.48e3T^{2} \)
61 \( 1 - 59.1T + 3.72e3T^{2} \)
71 \( 1 - 78.2iT - 5.04e3T^{2} \)
73 \( 1 + 54.9T + 5.32e3T^{2} \)
79 \( 1 + 42.1T + 6.24e3T^{2} \)
83 \( 1 + 4.52iT - 6.88e3T^{2} \)
89 \( 1 + 1.91iT - 7.92e3T^{2} \)
97 \( 1 - 83.7T + 9.40e3T^{2} \)
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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

−13.01804379101395047282137787230, −11.65295507858699348506389001378, −9.613700250125809641319429184115, −8.584695724591611749792201216051, −8.272622168303017969921132344266, −7.54707517157556672402028501518, −6.11417563744480454719485730380, −5.11805555202114078862552767122, −4.10881331002141287417424565183, −1.09321063925198284494873879930, 2.04139785400487546050666695738, 2.71549706234010123263110512714, 4.22936802966465164444105817709, 4.86468924854245519450854893719, 7.47309521597222283156788235904, 8.498396994946224199204034835826, 9.589984371983945186533236348826, 10.43123600357423489095796732374, 10.92674792612432887701423991946, 11.83392989251946858626441792948

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