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Properties

Label 2-201-201.17-c2-0-27
Degree $2$
Conductor $201$
Sign $0.371 + 0.928i$
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  + (−2.52 + 1.62i)3-s + (3.92 − 0.757i)4-s + (−6.83 − 9.59i)7-s + (3.73 − 8.18i)9-s + (−8.68 + 8.28i)12-s + (−0.347 − 1.43i)13-s + (14.8 − 5.94i)16-s + (21.3 − 30.0i)19-s + (32.8 + 13.1i)21-s + (−23.9 − 7.04i)25-s + (3.84 + 26.7i)27-s + (−34.1 − 32.5i)28-s + (14.5 − 60.1i)31-s + (8.48 − 34.9i)36-s + (22.2 + 38.5i)37-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)3-s + (0.981 − 0.189i)4-s + (−0.976 − 1.37i)7-s + (0.415 − 0.909i)9-s + (−0.723 + 0.690i)12-s + (−0.0267 − 0.110i)13-s + (0.928 − 0.371i)16-s + (1.12 − 1.58i)19-s + (1.56 + 0.625i)21-s + (−0.959 − 0.281i)25-s + (0.142 + 0.989i)27-s + (−1.21 − 1.16i)28-s + (0.470 − 1.93i)31-s + (0.235 − 0.971i)36-s + (0.600 + 1.04i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.371 + 0.928i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.963404 - 0.652251i\)
\(L(\frac12)\) \(\approx\) \(0.963404 - 0.652251i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.52 - 1.62i)T \)
67 \( 1 + (-63.2 + 22.0i)T \)
good2 \( 1 + (-3.92 + 0.757i)T^{2} \)
5 \( 1 + (23.9 + 7.04i)T^{2} \)
7 \( 1 + (6.83 + 9.59i)T + (-16.0 + 46.3i)T^{2} \)
11 \( 1 + (-28.5 - 117. i)T^{2} \)
13 \( 1 + (0.347 + 1.43i)T + (-150. + 77.4i)T^{2} \)
17 \( 1 + (-268. - 107. i)T^{2} \)
19 \( 1 + (-21.3 + 30.0i)T + (-118. - 341. i)T^{2} \)
23 \( 1 + (-306. - 430. i)T^{2} \)
29 \( 1 + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-14.5 + 60.1i)T + (-854. - 440. i)T^{2} \)
37 \( 1 + (-22.2 - 38.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (1.32e3 - 1.03e3i)T^{2} \)
43 \( 1 + (27.1 - 31.3i)T + (-263. - 1.83e3i)T^{2} \)
47 \( 1 + (2.19e3 - 209. i)T^{2} \)
53 \( 1 + (399. - 2.78e3i)T^{2} \)
59 \( 1 + (-2.92e3 + 1.88e3i)T^{2} \)
61 \( 1 + (78.9 - 62.0i)T + (877. - 3.61e3i)T^{2} \)
71 \( 1 + (-4.67e3 + 1.87e3i)T^{2} \)
73 \( 1 + (6.91 - 5.43i)T + (1.25e3 - 5.17e3i)T^{2} \)
79 \( 1 + (72.9 - 69.5i)T + (296. - 6.23e3i)T^{2} \)
83 \( 1 + (-4.98e3 + 4.75e3i)T^{2} \)
89 \( 1 + (-3.29e3 - 7.20e3i)T^{2} \)
97 \( 1 + (43.5 + 75.4i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.66349013737007416058247754576, −11.16125900791905675621178389439, −10.05997536081680061456204409817, −9.649495404081302360878485747360, −7.57958958353267913808913443919, −6.76211314897147990983936381963, −5.91272778236601726802601463846, −4.43352520327755287619446136137, −3.12236923399292653046505888944, −0.72866326343227662894284078974, 1.84230372802242121773320224425, 3.25321834829364563006604881284, 5.49049634420772431070288506520, 6.10891785664742952372411983223, 7.09481011864102683369074419717, 8.175024284896828967898578232126, 9.621053504718710204347377428561, 10.60144369325329567071110623861, 11.86432197466126161389784767482, 12.10162490464934120727155927799

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