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Properties

Label 2-201-201.110-c3-0-15
Degree $2$
Conductor $201$
Sign $0.613 - 0.790i$
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  + (−2.80 − 4.37i)3-s + (5.23 + 6.04i)4-s + (4.82 + 2.20i)7-s + (−11.2 + 24.5i)9-s + (11.7 − 39.8i)12-s + (−10.6 + 36.2i)13-s + (−9.10 + 63.3i)16-s + (15.4 + 33.8i)19-s + (−3.92 − 27.2i)21-s + (119. + 35.2i)25-s + (138. − 19.9i)27-s + (11.9 + 40.6i)28-s + (92.1 + 313. i)31-s + (−207. + 60.8i)36-s + 228.·37-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)3-s + (0.654 + 0.755i)4-s + (0.260 + 0.118i)7-s + (−0.415 + 0.909i)9-s + (0.281 − 0.959i)12-s + (−0.227 + 0.773i)13-s + (−0.142 + 0.989i)16-s + (0.186 + 0.408i)19-s + (−0.0407 − 0.283i)21-s + (0.959 + 0.281i)25-s + (0.989 − 0.142i)27-s + (0.0806 + 0.274i)28-s + (0.533 + 1.81i)31-s + (−0.959 + 0.281i)36-s + 1.01·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.790i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.613 - 0.790i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ 0.613 - 0.790i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.37430 + 0.673126i\)
\(L(\frac12)\) \(\approx\) \(1.37430 + 0.673126i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.80 + 4.37i)T \)
67 \( 1 + (514. - 190. i)T \)
good2 \( 1 + (-5.23 - 6.04i)T^{2} \)
5 \( 1 + (-119. - 35.2i)T^{2} \)
7 \( 1 + (-4.82 - 2.20i)T + (224. + 259. i)T^{2} \)
11 \( 1 + (-1.27e3 - 374. i)T^{2} \)
13 \( 1 + (10.6 - 36.2i)T + (-1.84e3 - 1.18e3i)T^{2} \)
17 \( 1 + (699. + 4.86e3i)T^{2} \)
19 \( 1 + (-15.4 - 33.8i)T + (-4.49e3 + 5.18e3i)T^{2} \)
23 \( 1 + (-5.05e3 + 1.10e4i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-92.1 - 313. i)T + (-2.50e4 + 1.61e4i)T^{2} \)
37 \( 1 - 228.T + 5.06e4T^{2} \)
41 \( 1 + (-9.80e3 - 6.82e4i)T^{2} \)
43 \( 1 + (-268. - 233. i)T + (1.13e4 + 7.86e4i)T^{2} \)
47 \( 1 + (-4.31e4 + 9.44e4i)T^{2} \)
53 \( 1 + (-2.11e4 + 1.47e5i)T^{2} \)
59 \( 1 + (-1.72e5 + 1.11e5i)T^{2} \)
61 \( 1 + (681. - 97.9i)T + (2.17e5 - 6.39e4i)T^{2} \)
71 \( 1 + (5.09e4 - 3.54e5i)T^{2} \)
73 \( 1 + (76.8 + 534. i)T + (-3.73e5 + 1.09e5i)T^{2} \)
79 \( 1 + (-224. + 765. i)T + (-4.14e5 - 2.66e5i)T^{2} \)
83 \( 1 + (5.48e5 + 1.61e5i)T^{2} \)
89 \( 1 + (-2.92e5 - 6.41e5i)T^{2} \)
97 \( 1 + 434. iT - 9.12e5T^{2} \)
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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

−12.13100311041045897258039751963, −11.40567456233782295884627983560, −10.54069428956910350587734585639, −8.899177035364451378884846629496, −7.87300257760019901541247969184, −7.02558917594227360942588594231, −6.15283883125555452876200063806, −4.70079138135097187886358538698, −2.92385081050622759866580516058, −1.55409744144064674399809167321, 0.73296003576768113360046697963, 2.75996366763110539872225104596, 4.45928133180692164132011782246, 5.51224578974682228347616810768, 6.40122201558181592763490290638, 7.70001293487681893472131906609, 9.201425490317984546049247532234, 10.07871319944508640692686070079, 10.86755933481753618496400814726, 11.53102487555947940846037311957

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