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Properties

Label 2-201-67.29-c3-0-13
Degree $2$
Conductor $201$
Sign $-0.220 - 0.975i$
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  + (−0.970 + 1.68i)2-s + 3·3-s + (2.11 + 3.66i)4-s + 11.3·5-s + (−2.91 + 5.04i)6-s + (14.7 + 25.6i)7-s − 23.7·8-s + 9·9-s + (−11.0 + 19.1i)10-s + (−1.65 − 2.86i)11-s + (6.34 + 10.9i)12-s + (3.34 − 5.79i)13-s − 57.4·14-s + 34.1·15-s + (6.12 − 10.6i)16-s + (29.4 − 50.9i)17-s + ⋯
L(s)  = 1  + (−0.343 + 0.594i)2-s + 0.577·3-s + (0.264 + 0.457i)4-s + 1.01·5-s + (−0.198 + 0.343i)6-s + (0.798 + 1.38i)7-s − 1.04·8-s + 0.333·9-s + (−0.349 + 0.604i)10-s + (−0.0453 − 0.0785i)11-s + (0.152 + 0.264i)12-s + (0.0713 − 0.123i)13-s − 1.09·14-s + 0.587·15-s + (0.0957 − 0.165i)16-s + (0.419 − 0.726i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.46439 + 1.83210i\)
\(L(\frac12)\) \(\approx\) \(1.46439 + 1.83210i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
67 \( 1 + (-411. - 362. i)T \)
good2 \( 1 + (0.970 - 1.68i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 11.3T + 125T^{2} \)
7 \( 1 + (-14.7 - 25.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (1.65 + 2.86i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-3.34 + 5.79i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-29.4 + 50.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-8.98 + 15.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-12.7 + 22.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-61.4 - 106. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (67.3 + 116. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (172. - 298. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (57.5 + 99.6i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 45.2T + 7.95e4T^{2} \)
47 \( 1 + (19.1 + 33.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 160.T + 1.48e5T^{2} \)
59 \( 1 - 383.T + 2.05e5T^{2} \)
61 \( 1 + (26.2 - 45.5i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (401. + 695. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (166. - 288. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (92.7 + 160. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-721. + 1.24e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 198.T + 7.04e5T^{2} \)
97 \( 1 + (-394. + 683. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.24591205419167277469693819465, −11.47731837883507087011394429502, −9.965500942170592864765633348362, −8.964837407199312254908707695343, −8.432361521169206909588127238709, −7.31327738063616567806738063262, −6.08867155614700616330833222031, −5.14041224014627875619883278426, −3.03473121680262054237365063675, −2.00436436811127759770000496875, 1.16070099881638034372618849761, 2.10962231447554325560677432055, 3.76414643373039534507601896809, 5.32413693906883731866365951910, 6.58852970447247423382008201355, 7.75928146095955364191510698784, 8.973534906530219716700139117413, 10.04539936826516699309373604463, 10.44756096547444371584146453547, 11.43294024485033023779136363975

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