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Properties

Label 2-201-67.38-c2-0-17
Degree $2$
Conductor $201$
Sign $0.936 + 0.349i$
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  + (1.57 + 0.908i)2-s + 1.73i·3-s + (−0.350 − 0.606i)4-s − 5.97i·5-s + (−1.57 + 2.72i)6-s + (6.73 − 3.89i)7-s − 8.53i·8-s − 2.99·9-s + (5.42 − 9.39i)10-s + (3.44 − 1.98i)11-s + (1.05 − 0.606i)12-s + (4.64 + 2.68i)13-s + 14.1·14-s + 10.3·15-s + (6.35 − 11.0i)16-s + (−6.53 + 11.3i)17-s + ⋯
L(s)  = 1  + (0.786 + 0.454i)2-s + 0.577i·3-s + (−0.0876 − 0.151i)4-s − 1.19i·5-s + (−0.262 + 0.454i)6-s + (0.962 − 0.555i)7-s − 1.06i·8-s − 0.333·9-s + (0.542 − 0.939i)10-s + (0.312 − 0.180i)11-s + (0.0876 − 0.0505i)12-s + (0.357 + 0.206i)13-s + 1.00·14-s + 0.689·15-s + (0.397 − 0.687i)16-s + (−0.384 + 0.665i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.936 + 0.349i$
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: \((2,\ 201,\ (\ :1),\ 0.936 + 0.349i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.23700 - 0.403906i\)
\(L(\frac12)\) \(\approx\) \(2.23700 - 0.403906i\)
\(L(2)\) 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 \( 1 + (-31.1 - 59.3i)T \)
good2 \( 1 + (-1.57 - 0.908i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + 5.97iT - 25T^{2} \)
7 \( 1 + (-6.73 + 3.89i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-3.44 + 1.98i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-4.64 - 2.68i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (6.53 - 11.3i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (6.51 - 11.2i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-14.7 + 25.5i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-16.8 - 29.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (24.0 - 13.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-2.99 + 5.18i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-30.7 + 17.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 35.0iT - 1.84e3T^{2} \)
47 \( 1 + (13.1 + 22.7i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 58.9iT - 2.80e3T^{2} \)
59 \( 1 + 47.4T + 3.48e3T^{2} \)
61 \( 1 + (-60.5 - 34.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
71 \( 1 + (-31.8 - 55.2i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (46.7 - 81.0i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-11.8 + 6.84i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (23.3 - 40.4i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 89.7T + 7.92e3T^{2} \)
97 \( 1 + (4.03 + 2.32i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.51891784176675992605195578542, −11.15963626779785315833263402748, −10.28114441170934743386852156110, −9.010171668244072919895041342898, −8.330087747004513323644673248886, −6.74754146332951713347246199775, −5.48298188502530854612971506450, −4.64871336440351458611275604859, −3.94247711003566716413047376546, −1.19059659827649774707793419993, 2.13556952960403747416783984809, 3.17375729846308795576338213444, 4.65038708686921272205104897141, 5.87822453870976197388479771932, 7.15071952096150663891338969009, 8.096005640470330327342121107036, 9.234993184570436467096453309192, 10.98494471495267350135896967232, 11.34283109602377673444635420708, 12.21507467335854383742005087404

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