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Properties

Label 2-201-201.164-c1-0-2
Degree $2$
Conductor $201$
Sign $0.234 + 0.972i$
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  + (−1.30 − 2.25i)2-s + (−1.69 − 0.341i)3-s + (−2.40 + 4.15i)4-s + 3.98·5-s + (1.44 + 4.28i)6-s + (2.31 + 1.33i)7-s + 7.30·8-s + (2.76 + 1.16i)9-s + (−5.19 − 9.00i)10-s + (−2.06 + 3.57i)11-s + (5.49 − 6.23i)12-s + (1.09 − 0.634i)13-s − 6.97i·14-s + (−6.76 − 1.36i)15-s + (−4.72 − 8.18i)16-s + (−1.45 + 0.841i)17-s + ⋯
L(s)  = 1  + (−0.922 − 1.59i)2-s + (−0.980 − 0.197i)3-s + (−1.20 + 2.07i)4-s + 1.78·5-s + (0.588 + 1.74i)6-s + (0.875 + 0.505i)7-s + 2.58·8-s + (0.922 + 0.387i)9-s + (−1.64 − 2.84i)10-s + (−0.621 + 1.07i)11-s + (1.58 − 1.80i)12-s + (0.305 − 0.176i)13-s − 1.86i·14-s + (−1.74 − 0.352i)15-s + (−1.18 − 2.04i)16-s + (−0.353 + 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.234 + 0.972i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.234 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591716 - 0.465899i\)
\(L(\frac12)\) \(\approx\) \(0.591716 - 0.465899i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.69 + 0.341i)T \)
67 \( 1 + (3.22 - 7.52i)T \)
good2 \( 1 + (1.30 + 2.25i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 3.98T + 5T^{2} \)
7 \( 1 + (-2.31 - 1.33i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.06 - 3.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.09 + 0.634i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.45 - 0.841i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.716 - 1.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.35 + 2.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.28 - 0.743i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.70 + 4.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0952 + 0.165i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.545 + 0.944i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.84iT - 43T^{2} \)
47 \( 1 + (-5.34 - 3.08i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.507T + 53T^{2} \)
59 \( 1 - 5.37iT - 59T^{2} \)
61 \( 1 + (-1.26 + 0.727i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (13.2 + 7.64i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.84 + 3.18i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.89 - 1.67i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.39 + 3.69i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
97 \( 1 + (8.00 - 4.61i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.11192161451128438508840720131, −10.97116769078260708654768717304, −10.46601445915796987512746910757, −9.643281144146677874556396641993, −8.753905108411900749990935812048, −7.33880833135458553435105932710, −5.72206830494557414323212507984, −4.67996760915378426705332865928, −2.35304032172713921580934971650, −1.55449293537080892984000159988, 1.24659665366132944765834124488, 4.98489209904424686504393215300, 5.55391503749285480208797088530, 6.41776102915370621934937462703, 7.35698574208739850138003189313, 8.731297956859830106798357483574, 9.509146984638698458180011273283, 10.53158837314813437474874883217, 11.05016595118863203319562846195, 13.15740748077937250624840491189

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