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Properties

Label 2-201-67.38-c2-0-10
Degree $2$
Conductor $201$
Sign $0.306 + 0.951i$
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  + (−2.11 − 1.21i)2-s + 1.73i·3-s + (0.970 + 1.68i)4-s − 1.83i·5-s + (2.11 − 3.65i)6-s + (4.69 − 2.71i)7-s + 5.02i·8-s − 2.99·9-s + (−2.23 + 3.87i)10-s + (−4.86 + 2.81i)11-s + (−2.91 + 1.68i)12-s + (5.62 + 3.25i)13-s − 13.2·14-s + 3.17·15-s + (9.99 − 17.3i)16-s + (2.30 − 3.99i)17-s + ⋯
L(s)  = 1  + (−1.05 − 0.609i)2-s + 0.577i·3-s + (0.242 + 0.420i)4-s − 0.367i·5-s + (0.351 − 0.609i)6-s + (0.670 − 0.387i)7-s + 0.627i·8-s − 0.333·9-s + (−0.223 + 0.387i)10-s + (−0.442 + 0.255i)11-s + (−0.242 + 0.140i)12-s + (0.433 + 0.250i)13-s − 0.943·14-s + 0.211·15-s + (0.624 − 1.08i)16-s + (0.135 − 0.234i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.306 + 0.951i$
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: \((2,\ 201,\ (\ :1),\ 0.306 + 0.951i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.689293 - 0.501932i\)
\(L(\frac12)\) \(\approx\) \(0.689293 - 0.501932i\)
\(L(2)\) 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 \( 1 + (27.2 - 61.2i)T \)
good2 \( 1 + (2.11 + 1.21i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + 1.83iT - 25T^{2} \)
7 \( 1 + (-4.69 + 2.71i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (4.86 - 2.81i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.62 - 3.25i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-2.30 + 3.99i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-14.1 + 24.5i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-10.1 + 17.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (17.8 + 30.9i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-2.44 + 1.41i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-25.9 + 44.9i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-15.3 + 8.86i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 0.469iT - 1.84e3T^{2} \)
47 \( 1 + (-4.50 - 7.79i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 31.8iT - 2.80e3T^{2} \)
59 \( 1 + 0.478T + 3.48e3T^{2} \)
61 \( 1 + (-66.7 - 38.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
71 \( 1 + (-57.4 - 99.4i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-11.2 + 19.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (42.8 - 24.7i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (8.02 - 13.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 27.0T + 7.92e3T^{2} \)
97 \( 1 + (83.9 + 48.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.41469762898945668108412214969, −11.04926521057553813886806635610, −10.00305732330380742843785596103, −9.173689520519966388355354691967, −8.384706924347703292434141779411, −7.29430492763300720631369124743, −5.42910095702788796281553328027, −4.43382739678309693404235978155, −2.55777155080503622337478921338, −0.813770880549581333602408509004, 1.34172560787413049279240832178, 3.33630691094314872669261888871, 5.38546537395971718811000372183, 6.51604743295253497415050142312, 7.63893463917607994367629207768, 8.188613303425961370920129406203, 9.175999155930040729812972115446, 10.32527419579324362196999225571, 11.28040986464426409160241977185, 12.41297216256402052693194923482

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