Properties

Label 2-201-67.66-c4-0-5
Degree $2$
Conductor $201$
Sign $-0.999 - 0.00587i$
Analytic cond. $20.7773$
Root an. cond. $4.55821$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.58i·2-s − 5.19i·3-s − 41.4·4-s − 47.0i·5-s + 39.3·6-s + 16.7i·7-s − 193. i·8-s − 27·9-s + 356.·10-s + 0.510i·11-s + 215. i·12-s + 290. i·13-s − 126.·14-s − 244.·15-s + 801.·16-s − 349.·17-s + ⋯
L(s)  = 1  + 1.89i·2-s − 0.577i·3-s − 2.59·4-s − 1.88i·5-s + 1.09·6-s + 0.341i·7-s − 3.01i·8-s − 0.333·9-s + 3.56·10-s + 0.00421i·11-s + 1.49i·12-s + 1.71i·13-s − 0.646·14-s − 1.08·15-s + 3.13·16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00587i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.999 - 0.00587i$
Analytic conductor: \(20.7773\)
Root analytic conductor: \(4.55821\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :2),\ -0.999 - 0.00587i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.7494493670\)
\(L(\frac12)\) \(\approx\) \(0.7494493670\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19iT \)
67 \( 1 + (-26.3 + 4.48e3i)T \)
good2 \( 1 - 7.58iT - 16T^{2} \)
5 \( 1 + 47.0iT - 625T^{2} \)
7 \( 1 - 16.7iT - 2.40e3T^{2} \)
11 \( 1 - 0.510iT - 1.46e4T^{2} \)
13 \( 1 - 290. iT - 2.85e4T^{2} \)
17 \( 1 + 349.T + 8.35e4T^{2} \)
19 \( 1 - 108.T + 1.30e5T^{2} \)
23 \( 1 - 481.T + 2.79e5T^{2} \)
29 \( 1 - 509.T + 7.07e5T^{2} \)
31 \( 1 - 1.34e3iT - 9.23e5T^{2} \)
37 \( 1 + 176.T + 1.87e6T^{2} \)
41 \( 1 - 2.76e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.92e3iT - 3.41e6T^{2} \)
47 \( 1 + 183.T + 4.87e6T^{2} \)
53 \( 1 + 386. iT - 7.89e6T^{2} \)
59 \( 1 - 1.49e3T + 1.21e7T^{2} \)
61 \( 1 + 650. iT - 1.38e7T^{2} \)
71 \( 1 + 8.62e3T + 2.54e7T^{2} \)
73 \( 1 + 7.85e3T + 2.83e7T^{2} \)
79 \( 1 - 5.67e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.40e3T + 4.74e7T^{2} \)
89 \( 1 + 1.11e4T + 6.27e7T^{2} \)
97 \( 1 - 9.40e3iT - 8.85e7T^{2} \)
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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

−12.71548895154086740527193295037, −11.72384812710455031890307022156, −9.421833629190173782079669606890, −8.885969836110138526644496125416, −8.301331639747642663422196338839, −7.06848172920591443844200673982, −6.19414788298150648140321420522, −4.98596071676073303640885690726, −4.42655852674055885968776249922, −1.25981890164029919574814130687, 0.29727640147498862251485821515, 2.43279165186858093608385829704, 3.15849683818273178765433973133, 4.11915337633351269770462525174, 5.64663042859958440195665727983, 7.35020282552464152352489580942, 8.749179176022593547261742925635, 9.988839020507093245330355703801, 10.52947333182947358132387282291, 10.98506422955515318370187703939

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