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Properties

Label 2-201-67.38-c2-0-15
Degree $2$
Conductor $201$
Sign $-0.894 + 0.447i$
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  + (−2.78 − 1.60i)2-s − 1.73i·3-s + (3.16 + 5.48i)4-s − 6.80i·5-s + (−2.78 + 4.82i)6-s + (9.81 − 5.66i)7-s − 7.49i·8-s − 2.99·9-s + (−10.9 + 18.9i)10-s + (14.2 − 8.24i)11-s + (9.49 − 5.48i)12-s + (14.8 + 8.57i)13-s − 36.4·14-s − 11.7·15-s + (0.620 − 1.07i)16-s + (11.5 − 20.0i)17-s + ⋯
L(s)  = 1  + (−1.39 − 0.803i)2-s − 0.577i·3-s + (0.791 + 1.37i)4-s − 1.36i·5-s + (−0.463 + 0.803i)6-s + (1.40 − 0.809i)7-s − 0.936i·8-s − 0.333·9-s + (−1.09 + 1.89i)10-s + (1.29 − 0.749i)11-s + (0.791 − 0.456i)12-s + (1.14 + 0.659i)13-s − 2.60·14-s − 0.785·15-s + (0.0387 − 0.0672i)16-s + (0.681 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.894 + 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.222011 - 0.939751i\)
\(L(\frac12)\) \(\approx\) \(0.222011 - 0.939751i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
67 \( 1 + (-64.6 - 17.6i)T \)
good2 \( 1 + (2.78 + 1.60i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + 6.80iT - 25T^{2} \)
7 \( 1 + (-9.81 + 5.66i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-14.2 + 8.24i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-14.8 - 8.57i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-11.5 + 20.0i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-1.69 + 2.92i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (20.3 - 35.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-3.45 - 5.98i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (32.4 - 18.7i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (13.3 - 23.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-16.0 + 9.26i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 0.773iT - 1.84e3T^{2} \)
47 \( 1 + (7.84 + 13.5i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 70.3iT - 2.80e3T^{2} \)
59 \( 1 + 5.22T + 3.48e3T^{2} \)
61 \( 1 + (15.8 + 9.13i)T + (1.86e3 + 3.22e3i)T^{2} \)
71 \( 1 + (-54.5 - 94.4i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (41.0 - 71.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-99.9 + 57.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (48.3 - 83.6i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 130.T + 7.92e3T^{2} \)
97 \( 1 + (49.6 + 28.6i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.51591957027164128036869956823, −11.11701571098903184406973089767, −9.566288225723129129267022030809, −8.798994013391795138543854997602, −8.178771978862435743043611253088, −7.19841112145622859484752361433, −5.38173757251745265938482868031, −3.83615015966862609213970603731, −1.41994470811479864800580131457, −1.12938718554916313416044026448, 1.82446011977355415465865427693, 3.89132541147281792145822788598, 5.79508270421853895216757845623, 6.57974943257198767155411728965, 7.892147270476021095524658455828, 8.494900995567283317427798748808, 9.571243596402426613156244867100, 10.60232921305336320613730698858, 11.05960219586739302427599377691, 12.27710329697968289408370461160

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