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Properties

Label 2-201-201.152-c1-0-3
Degree $2$
Conductor $201$
Sign $-0.426 - 0.904i$
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 + (0.654 + 1.89i)4-s + (−0.0717 + 0.751i)7-s + (−1.24 − 2.72i)9-s + (−3.36 − 0.816i)12-s + (−4.89 + 5.12i)13-s + (−3.14 + 2.47i)16-s + (8.56 − 0.817i)19-s + (−1.02 − 0.807i)21-s + (4.79 − 1.40i)25-s + (5.14 + 0.739i)27-s + (−1.46 + 0.355i)28-s + (−5.63 − 5.90i)31-s + (4.34 − 4.14i)36-s + (5.77 + 9.99i)37-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)3-s + (0.327 + 0.945i)4-s + (−0.0271 + 0.283i)7-s + (−0.415 − 0.909i)9-s + (−0.971 − 0.235i)12-s + (−1.35 + 1.42i)13-s + (−0.786 + 0.618i)16-s + (1.96 − 0.187i)19-s + (−0.224 − 0.176i)21-s + (0.959 − 0.281i)25-s + (0.989 + 0.142i)27-s + (−0.277 + 0.0672i)28-s + (−1.01 − 1.06i)31-s + (0.723 − 0.690i)36-s + (0.949 + 1.64i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.426 - 0.904i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ -0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522801 + 0.824157i\)
\(L(\frac12)\) \(\approx\) \(0.522801 + 0.824157i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.202 + 8.18i)T \)
good2 \( 1 + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (-4.79 + 1.40i)T^{2} \)
7 \( 1 + (0.0717 - 0.751i)T + (-6.87 - 1.32i)T^{2} \)
11 \( 1 + (7.96 + 7.59i)T^{2} \)
13 \( 1 + (4.89 - 5.12i)T + (-0.618 - 12.9i)T^{2} \)
17 \( 1 + (13.3 + 10.5i)T^{2} \)
19 \( 1 + (-8.56 + 0.817i)T + (18.6 - 3.59i)T^{2} \)
23 \( 1 + (22.8 + 2.18i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.63 + 5.90i)T + (-1.47 + 30.9i)T^{2} \)
37 \( 1 + (-5.77 - 9.99i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (38.0 - 15.2i)T^{2} \)
43 \( 1 + (-7.72 + 6.69i)T + (6.11 - 42.5i)T^{2} \)
47 \( 1 + (-27.2 + 38.2i)T^{2} \)
53 \( 1 + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (0.0955 + 0.238i)T + (-44.1 + 42.0i)T^{2} \)
71 \( 1 + (55.8 - 43.8i)T^{2} \)
73 \( 1 + (-0.654 + 0.261i)T + (52.8 - 50.3i)T^{2} \)
79 \( 1 + (-9.49 - 2.30i)T + (70.2 + 36.1i)T^{2} \)
83 \( 1 + (-19.5 + 80.6i)T^{2} \)
89 \( 1 + (-36.9 + 80.9i)T^{2} \)
97 \( 1 + (12.6 - 7.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.30252999582439799512223577987, −11.85349540025136270020976775881, −11.04450671460682400665636163716, −9.659017348657145812062147951553, −9.076506563631324519513421480792, −7.59922094841661772123930248500, −6.65738160389713313471416775560, −5.19695254614827497564801331085, −4.11605976504231210725203257488, −2.74272804982126842085098075459, 0.946152833779534862610505785133, 2.71420523717421541933314395080, 5.12466983370408740885774435481, 5.69373463179608559335358949351, 7.12199238969762991439408737796, 7.63718149121226719643308335479, 9.368427441525001372689303322851, 10.38383806883071039654805949178, 11.11600245111964618360980791754, 12.18989119528831607409937931385

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