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Properties

Label 2-201-201.122-c2-0-27
Degree $2$
Conductor $201$
Sign $0.903 + 0.428i$
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  + (1.96 − 2.26i)3-s + (2.89 + 2.76i)4-s + (4.57 − 3.59i)7-s + (−1.28 − 8.90i)9-s + (11.9 − 1.14i)12-s + (5.06 + 7.10i)13-s + (0.761 + 15.9i)16-s + (−3.07 − 2.42i)19-s + (0.831 − 17.4i)21-s + (10.3 − 22.7i)25-s + (−22.7 − 14.5i)27-s + (23.1 + 2.21i)28-s + (12.6 − 17.7i)31-s + (20.8 − 29.3i)36-s + (−36.9 + 63.9i)37-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)3-s + (0.723 + 0.690i)4-s + (0.653 − 0.514i)7-s + (−0.142 − 0.989i)9-s + (0.995 − 0.0950i)12-s + (0.389 + 0.546i)13-s + (0.0475 + 0.998i)16-s + (−0.162 − 0.127i)19-s + (0.0395 − 0.830i)21-s + (0.415 − 0.909i)25-s + (−0.841 − 0.540i)27-s + (0.828 + 0.0790i)28-s + (0.408 − 0.573i)31-s + (0.580 − 0.814i)36-s + (−0.997 + 1.72i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.903 + 0.428i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.903 + 0.428i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.22093 - 0.500305i\)
\(L(\frac12)\) \(\approx\) \(2.22093 - 0.500305i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.96 + 2.26i)T \)
67 \( 1 + (58.0 - 33.3i)T \)
good2 \( 1 + (-2.89 - 2.76i)T^{2} \)
5 \( 1 + (-10.3 + 22.7i)T^{2} \)
7 \( 1 + (-4.57 + 3.59i)T + (11.5 - 47.6i)T^{2} \)
11 \( 1 + (-70.1 - 98.5i)T^{2} \)
13 \( 1 + (-5.06 - 7.10i)T + (-55.2 + 159. i)T^{2} \)
17 \( 1 + (-13.7 + 288. i)T^{2} \)
19 \( 1 + (3.07 + 2.42i)T + (85.1 + 350. i)T^{2} \)
23 \( 1 + (415. - 327. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-12.6 + 17.7i)T + (-314. - 908. i)T^{2} \)
37 \( 1 + (36.9 - 63.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (1.49e3 - 770. i)T^{2} \)
43 \( 1 + (28.5 + 8.39i)T + (1.55e3 + 9.99e2i)T^{2} \)
47 \( 1 + (-2.05e3 - 821. i)T^{2} \)
53 \( 1 + (-2.36e3 + 1.51e3i)T^{2} \)
59 \( 1 + (2.27e3 - 2.63e3i)T^{2} \)
61 \( 1 + (48.2 - 24.8i)T + (2.15e3 - 3.03e3i)T^{2} \)
71 \( 1 + (-239. - 5.03e3i)T^{2} \)
73 \( 1 + (121. - 62.4i)T + (3.09e3 - 4.34e3i)T^{2} \)
79 \( 1 + (-25.8 + 2.46i)T + (6.12e3 - 1.18e3i)T^{2} \)
83 \( 1 + (6.85e3 - 654. i)T^{2} \)
89 \( 1 + (1.12e3 - 7.84e3i)T^{2} \)
97 \( 1 + (-91.9 + 159. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.09065991490794943278375952368, −11.46815126135976722385140448123, −10.29953872718558293801977020534, −8.771111659374341917519886654449, −8.068424733300472781955048436026, −7.13356839909183538511605745069, −6.30665431954936751583005854459, −4.29451523518121364634986884549, −2.95882189096967002047157503327, −1.59342509524605268294879027383, 1.85651616025767959437945609766, 3.23418829690397668806159388138, 4.89282775406505566935206910103, 5.79332839378630361577182256010, 7.29448489724166711928664845759, 8.415151368995098618181711688012, 9.345156645686107224391346135634, 10.47532822787702262635315625638, 11.01787090392301580555717011253, 12.11000825847359057508598691841

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