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Properties

Label 2-201-67.38-c2-0-2
Degree $2$
Conductor $201$
Sign $-0.994 + 0.101i$
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  + (−2.55 − 1.47i)2-s + 1.73i·3-s + (2.34 + 4.06i)4-s + 9.46i·5-s + (2.55 − 4.42i)6-s + (−6.24 + 3.60i)7-s − 2.05i·8-s − 2.99·9-s + (13.9 − 24.1i)10-s + (0.365 − 0.211i)11-s + (−7.04 + 4.06i)12-s + (−15.7 − 9.11i)13-s + 21.2·14-s − 16.3·15-s + (6.36 − 11.0i)16-s + (12.3 − 21.3i)17-s + ⋯
L(s)  = 1  + (−1.27 − 0.737i)2-s + 0.577i·3-s + (0.587 + 1.01i)4-s + 1.89i·5-s + (0.425 − 0.737i)6-s + (−0.891 + 0.514i)7-s − 0.256i·8-s − 0.333·9-s + (1.39 − 2.41i)10-s + (0.0332 − 0.0192i)11-s + (−0.587 + 0.338i)12-s + (−1.21 − 0.701i)13-s + 1.51·14-s − 1.09·15-s + (0.397 − 0.689i)16-s + (0.724 − 1.25i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.994 + 0.101i$
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: \((2,\ 201,\ (\ :1),\ -0.994 + 0.101i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00968586 - 0.190205i\)
\(L(\frac12)\) \(\approx\) \(0.00968586 - 0.190205i\)
\(L(2)\) 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 \( 1 + (54.1 + 39.4i)T \)
good2 \( 1 + (2.55 + 1.47i)T + (2 + 3.46i)T^{2} \)
5 \( 1 - 9.46iT - 25T^{2} \)
7 \( 1 + (6.24 - 3.60i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-0.365 + 0.211i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (15.7 + 9.11i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (-12.3 + 21.3i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-9.32 + 16.1i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (11.8 - 20.4i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-16.8 - 29.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-10.8 + 6.24i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-7.83 + 13.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (21.5 - 12.4i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 - 10.0iT - 1.84e3T^{2} \)
47 \( 1 + (22.6 + 39.2i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 63.1iT - 2.80e3T^{2} \)
59 \( 1 + 7.36T + 3.48e3T^{2} \)
61 \( 1 + (-60.6 - 34.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
71 \( 1 + (56.6 + 98.1i)T + (-2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (39.6 - 68.6i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (2.84 - 1.64i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (3.68 - 6.37i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 90.8T + 7.92e3T^{2} \)
97 \( 1 + (160. + 92.7i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.10287741536595087753796650301, −11.45004174108239124542944737043, −10.46469156224288763434700621216, −9.863412874831869285234624985835, −9.329730470919849326820597657228, −7.73868671018376002010142137091, −6.95086776705004078596195713651, −5.46730409718844995203573130531, −3.06399434344610121582695419963, −2.75160421270773684062580610156, 0.16681636251623752121348893334, 1.46772901279377052813663261371, 4.18140727565842659387304057444, 5.73496163930654067248918938329, 6.78722759188841116001904441375, 7.983185015782646881167775265252, 8.451784513537829602701051922605, 9.667635522192853560703556747579, 10.01920616983852178852302195634, 12.05435393539600621656940531440

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