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Properties

Label 2-201-67.29-c3-0-0
Degree $2$
Conductor $201$
Sign $-0.317 - 0.948i$
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  + (1.20 − 2.09i)2-s − 3·3-s + (1.08 + 1.87i)4-s − 7.44·5-s + (−3.62 + 6.27i)6-s + (−10.2 − 17.8i)7-s + 24.5·8-s + 9·9-s + (−8.98 + 15.5i)10-s + (−6.01 − 10.4i)11-s + (−3.24 − 5.62i)12-s + (−26.9 + 46.6i)13-s − 49.7·14-s + 22.3·15-s + (20.9 − 36.3i)16-s + (−37.4 + 64.9i)17-s + ⋯
L(s)  = 1  + (0.426 − 0.739i)2-s − 0.577·3-s + (0.135 + 0.234i)4-s − 0.665·5-s + (−0.246 + 0.426i)6-s + (−0.555 − 0.962i)7-s + 1.08·8-s + 0.333·9-s + (−0.284 + 0.492i)10-s + (−0.164 − 0.285i)11-s + (−0.0781 − 0.135i)12-s + (−0.574 + 0.994i)13-s − 0.949·14-s + 0.384·15-s + (0.327 − 0.567i)16-s + (−0.534 + 0.926i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.317 - 0.948i$
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: \((2,\ 201,\ (\ :3/2),\ -0.317 - 0.948i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.246570 + 0.342442i\)
\(L(\frac12)\) \(\approx\) \(0.246570 + 0.342442i\)
\(L(\frac{5}{2})\) 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 \( 1 + (446. + 319. i)T \)
good2 \( 1 + (-1.20 + 2.09i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 7.44T + 125T^{2} \)
7 \( 1 + (10.2 + 17.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (6.01 + 10.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (26.9 - 46.6i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (37.4 - 64.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (67.7 - 117. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (77.3 - 134. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (63.8 + 110. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (39.0 + 67.6i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-192. + 333. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-65.3 - 113. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + 414.T + 7.95e4T^{2} \)
47 \( 1 + (-86.8 - 150. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 561.T + 1.48e5T^{2} \)
59 \( 1 + 259.T + 2.05e5T^{2} \)
61 \( 1 + (45.1 - 78.2i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (-250. - 433. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-92.7 + 160. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (263. + 455. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-58.8 + 101. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 903.T + 7.04e5T^{2} \)
97 \( 1 + (183. - 317. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.14719477989156529400348612401, −11.45542208464756409206840836374, −10.62919199593421552060101275973, −9.758184416987994311928312637538, −8.013716582068653914864848037277, −7.23993618407334830468018816224, −6.01723337124611711283114669720, −4.14379916025563495639724541170, −3.85025016885562706497404826263, −1.87822142839203509563491904117, 0.15951962936369713239433264540, 2.55870140634317827911425108706, 4.52673421937929189079806977456, 5.37146566876802248723427848596, 6.46695198015026459644355820022, 7.25504592751294728658274433513, 8.495307908619862548898613235067, 9.837220315354093788174127032571, 10.81220157369498916381637002496, 11.80338462057871527173886204477

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