Properties

Label 2-201-67.30-c2-0-19
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: $\chi_{201} (97, \cdot )$
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

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

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