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Properties

Label 2-201-201.50-c1-0-4
Degree $2$
Conductor $201$
Sign $-0.525 - 0.851i$
Analytic cond. $1.60499$
Root an. cond. $1.26688$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.936 + 1.45i)3-s + (−1.96 + 0.378i)4-s + (−4.31 + 3.06i)7-s + (−1.24 + 2.72i)9-s + (−2.39 − 2.50i)12-s + (5.48 − 1.33i)13-s + (3.71 − 1.48i)16-s + (−2.81 + 3.95i)19-s + (−8.50 − 3.40i)21-s + (4.79 + 1.40i)25-s + (−5.14 + 0.739i)27-s + (7.30 − 7.65i)28-s + (5.53 + 1.34i)31-s + (1.41 − 5.83i)36-s + (3.82 + 6.61i)37-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)3-s + (−0.981 + 0.189i)4-s + (−1.62 + 1.16i)7-s + (−0.415 + 0.909i)9-s + (−0.690 − 0.723i)12-s + (1.52 − 0.369i)13-s + (0.928 − 0.371i)16-s + (−0.646 + 0.907i)19-s + (−1.85 − 0.743i)21-s + (0.959 + 0.281i)25-s + (−0.989 + 0.142i)27-s + (1.38 − 1.44i)28-s + (0.994 + 0.241i)31-s + (0.235 − 0.971i)36-s + (0.628 + 1.08i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.525 - 0.851i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ -0.525 - 0.851i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433831 + 0.777368i\)
\(L(\frac12)\) \(\approx\) \(0.433831 + 0.777368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.936 - 1.45i)T \)
67 \( 1 + (-4.59 + 6.77i)T \)
good2 \( 1 + (1.96 - 0.378i)T^{2} \)
5 \( 1 + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (4.31 - 3.06i)T + (2.28 - 6.61i)T^{2} \)
11 \( 1 + (2.59 + 10.6i)T^{2} \)
13 \( 1 + (-5.48 + 1.33i)T + (11.5 - 5.95i)T^{2} \)
17 \( 1 + (-15.7 - 6.31i)T^{2} \)
19 \( 1 + (2.81 - 3.95i)T + (-6.21 - 17.9i)T^{2} \)
23 \( 1 + (-13.3 - 18.7i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.53 - 1.34i)T + (27.5 + 14.2i)T^{2} \)
37 \( 1 + (-3.82 - 6.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-32.2 + 25.3i)T^{2} \)
43 \( 1 + (9.85 + 8.54i)T + (6.11 + 42.5i)T^{2} \)
47 \( 1 + (46.7 - 4.46i)T^{2} \)
53 \( 1 + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-8.84 - 11.2i)T + (-14.3 + 59.2i)T^{2} \)
71 \( 1 + (-65.9 + 26.3i)T^{2} \)
73 \( 1 + (3.24 - 2.55i)T + (17.2 - 70.9i)T^{2} \)
79 \( 1 + (-0.135 - 0.142i)T + (-3.75 + 78.9i)T^{2} \)
83 \( 1 + (-60.0 + 57.2i)T^{2} \)
89 \( 1 + (-36.9 - 80.9i)T^{2} \)
97 \( 1 + (-5.11 + 2.95i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.05670117051718188776400804362, −11.97640706095786694880693901626, −10.42493330657160909307114747484, −9.747466344085668591982722062941, −8.742442217294377977858937939889, −8.401160890643942483902383809849, −6.34322141980179073741773215672, −5.33296554033988672919060913981, −3.83513979038020147607000558111, −3.02490753830077294844148072035, 0.77993728840844051776072943383, 3.22213868484916626896349666426, 4.20921912250857822847394796179, 6.20576976458001039035566623483, 6.83083385186756142655343405731, 8.224913499573992109237515733019, 9.100244786386865528252002012985, 9.935593592380714525843147303913, 11.07242375833650599647280678339, 12.67899064057113630146764481400

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