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Properties

Label 2-201-3.2-c2-0-17
Degree $2$
Conductor $201$
Sign $-0.0358 - 0.999i$
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  − 0.0529i·2-s + (0.107 + 2.99i)3-s + 3.99·4-s + 8.60i·5-s + (0.158 − 0.00570i)6-s + 10.6·7-s − 0.423i·8-s + (−8.97 + 0.645i)9-s + 0.455·10-s − 15.5i·11-s + (0.430 + 11.9i)12-s − 20.3·13-s − 0.563i·14-s + (−25.7 + 0.925i)15-s + 15.9·16-s + 2.77i·17-s + ⋯
L(s)  = 1  − 0.0264i·2-s + (0.0358 + 0.999i)3-s + 0.999·4-s + 1.72i·5-s + (0.0264 − 0.000950i)6-s + 1.51·7-s − 0.0529i·8-s + (−0.997 + 0.0716i)9-s + 0.0455·10-s − 1.41i·11-s + (0.0358 + 0.998i)12-s − 1.56·13-s − 0.0402i·14-s + (−1.71 + 0.0617i)15-s + 0.997·16-s + 0.163i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0358 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0358 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.0358 - 0.999i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.0358 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.40793 + 1.45938i\)
\(L(\frac12)\) \(\approx\) \(1.40793 + 1.45938i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.107 - 2.99i)T \)
67 \( 1 + 8.18T \)
good2 \( 1 + 0.0529iT - 4T^{2} \)
5 \( 1 - 8.60iT - 25T^{2} \)
7 \( 1 - 10.6T + 49T^{2} \)
11 \( 1 + 15.5iT - 121T^{2} \)
13 \( 1 + 20.3T + 169T^{2} \)
17 \( 1 - 2.77iT - 289T^{2} \)
19 \( 1 - 18.4T + 361T^{2} \)
23 \( 1 + 10.6iT - 529T^{2} \)
29 \( 1 + 8.83iT - 841T^{2} \)
31 \( 1 - 30.0T + 961T^{2} \)
37 \( 1 + 37.5T + 1.36e3T^{2} \)
41 \( 1 - 56.0iT - 1.68e3T^{2} \)
43 \( 1 + 10.9T + 1.84e3T^{2} \)
47 \( 1 + 36.0iT - 2.20e3T^{2} \)
53 \( 1 - 12.3iT - 2.80e3T^{2} \)
59 \( 1 + 87.1iT - 3.48e3T^{2} \)
61 \( 1 - 40.5T + 3.72e3T^{2} \)
71 \( 1 - 46.7iT - 5.04e3T^{2} \)
73 \( 1 + 59.7T + 5.32e3T^{2} \)
79 \( 1 - 11.5T + 6.24e3T^{2} \)
83 \( 1 + 112. iT - 6.88e3T^{2} \)
89 \( 1 - 26.6iT - 7.92e3T^{2} \)
97 \( 1 + 7.09T + 9.40e3T^{2} \)
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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

−11.61501334520205271331769624463, −11.54963295295199009267623607643, −10.59145370212009494497132053212, −9.998222336719720765996409194380, −8.277526588956827042475976398074, −7.42726470205497021288829190814, −6.22879996678498195482489039655, −5.06480709949577895369676044879, −3.35010171575245119364428427922, −2.44963166778882353283703832439, 1.31741958709050512416249360241, 2.18635028501209975155747671709, 4.77969963039201104303763985203, 5.42560626002181079637696165841, 7.26398618649621615649552536733, 7.65467983889077765834786267966, 8.713225141887405744082726916891, 9.948660945144108959327676516163, 11.50831290594543083822916252665, 12.17896187252793711126940732794

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