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Properties

Label 2-201-201.137-c3-0-54
Degree $2$
Conductor $201$
Sign $-0.739 - 0.672i$
Analytic cond. $11.8593$
Root an. cond. $3.44374$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.14 − 0.739i)3-s + (−6.73 − 4.32i)4-s + (10.4 − 35.5i)7-s + (25.9 + 7.60i)9-s + (31.4 + 27.2i)12-s + (−30.3 − 26.2i)13-s + (26.5 + 58.2i)16-s + (−155. + 45.7i)19-s + (−79.9 + 175. i)21-s + (81.8 − 94.4i)25-s + (−127. − 58.2i)27-s + (−223. + 194. i)28-s + (−260. + 225. i)31-s + (−141. − 163. i)36-s + 401.·37-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)3-s + (−0.841 − 0.540i)4-s + (0.563 − 1.91i)7-s + (0.959 + 0.281i)9-s + (0.755 + 0.654i)12-s + (−0.646 − 0.560i)13-s + (0.415 + 0.909i)16-s + (−1.87 + 0.551i)19-s + (−0.830 + 1.81i)21-s + (0.654 − 0.755i)25-s + (−0.909 − 0.415i)27-s + (−1.51 + 1.30i)28-s + (−1.51 + 1.30i)31-s + (−0.654 − 0.755i)36-s + 1.78·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.672i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.739 - 0.672i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.739 - 0.672i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0842896 + 0.218016i\)
\(L(\frac12)\) \(\approx\) \(0.0842896 + 0.218016i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.14 + 0.739i)T \)
67 \( 1 + (535. - 118. i)T \)
good2 \( 1 + (6.73 + 4.32i)T^{2} \)
5 \( 1 + (-81.8 + 94.4i)T^{2} \)
7 \( 1 + (-10.4 + 35.5i)T + (-288. - 185. i)T^{2} \)
11 \( 1 + (-871. + 1.00e3i)T^{2} \)
13 \( 1 + (30.3 + 26.2i)T + (312. + 2.17e3i)T^{2} \)
17 \( 1 + (-2.04e3 + 4.46e3i)T^{2} \)
19 \( 1 + (155. - 45.7i)T + (5.77e3 - 3.70e3i)T^{2} \)
23 \( 1 + (1.16e4 + 3.42e3i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (260. - 225. i)T + (4.23e3 - 2.94e4i)T^{2} \)
37 \( 1 - 401.T + 5.06e4T^{2} \)
41 \( 1 + (2.86e4 - 6.26e4i)T^{2} \)
43 \( 1 + (-289. - 450. i)T + (-3.30e4 + 7.23e4i)T^{2} \)
47 \( 1 + (9.96e4 + 2.92e4i)T^{2} \)
53 \( 1 + (6.18e4 + 1.35e5i)T^{2} \)
59 \( 1 + (2.92e4 - 2.03e5i)T^{2} \)
61 \( 1 + (284. + 130. i)T + (1.48e5 + 1.71e5i)T^{2} \)
71 \( 1 + (-1.48e5 - 3.25e5i)T^{2} \)
73 \( 1 + (346. - 759. i)T + (-2.54e5 - 2.93e5i)T^{2} \)
79 \( 1 + (-35.1 - 30.4i)T + (7.01e4 + 4.88e5i)T^{2} \)
83 \( 1 + (3.74e5 - 4.32e5i)T^{2} \)
89 \( 1 + (6.76e5 - 1.98e5i)T^{2} \)
97 \( 1 + 1.51e3iT - 9.12e5T^{2} \)
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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

−10.95426803924573324432831131196, −10.61456870215258298879031423790, −9.792568706081986787250387582036, −8.212495701075843480164636611854, −7.20080864002865422612391973835, −6.07174023184593806899498616406, −4.72656487615622561176992874184, −4.17714265685169983564964409816, −1.28896032737097182130050409148, −0.12866323792005689870661749180, 2.27909549878923813820335644152, 4.27076385675264990289833390167, 5.17024538708800109486526874342, 6.11545203283562747994220776915, 7.59340951485878884565291343645, 8.913342532862370356107907562030, 9.333320721972645877160909251894, 10.88523219438024402194330900741, 11.76080856435236308202200511963, 12.50409276516947885284422352526

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