Properties

Label 2-201-67.62-c1-0-11
Degree $2$
Conductor $201$
Sign $-0.782 + 0.623i$
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  + (1.65 − 1.90i)2-s + (−0.415 − 0.909i)3-s + (−0.623 − 4.33i)4-s + (−1.11 + 0.327i)5-s + (−2.42 − 0.711i)6-s + (−1.12 + 1.30i)7-s + (−5.06 − 3.25i)8-s + (−0.654 + 0.755i)9-s + (−1.21 + 2.66i)10-s + (3.48 − 1.02i)11-s + (−3.68 + 2.37i)12-s + (4.94 − 3.17i)13-s + (0.618 + 4.30i)14-s + (0.760 + 0.877i)15-s + (−6.19 + 1.81i)16-s + (−0.417 + 2.90i)17-s + ⋯
L(s)  = 1  + (1.16 − 1.35i)2-s + (−0.239 − 0.525i)3-s + (−0.311 − 2.16i)4-s + (−0.498 + 0.146i)5-s + (−0.989 − 0.290i)6-s + (−0.425 + 0.491i)7-s + (−1.79 − 1.15i)8-s + (−0.218 + 0.251i)9-s + (−0.385 + 0.844i)10-s + (1.05 − 0.308i)11-s + (−1.06 + 0.684i)12-s + (1.37 − 0.880i)13-s + (0.165 + 1.14i)14-s + (0.196 + 0.226i)15-s + (−1.54 + 0.454i)16-s + (−0.101 + 0.704i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.782 + 0.623i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ -0.782 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.592751 - 1.69554i\)
\(L(\frac12)\) \(\approx\) \(0.592751 - 1.69554i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (-4.19 + 7.03i)T \)
good2 \( 1 + (-1.65 + 1.90i)T + (-0.284 - 1.97i)T^{2} \)
5 \( 1 + (1.11 - 0.327i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (1.12 - 1.30i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (-3.48 + 1.02i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-4.94 + 3.17i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (0.417 - 2.90i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-2.18 - 2.52i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (1.30 + 2.85i)T + (-15.0 + 17.3i)T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + (-5.54 - 3.56i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 - 1.00T + 37T^{2} \)
41 \( 1 + (0.177 - 1.23i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (1.31 - 9.15i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 + (-0.0864 - 0.189i)T + (-30.7 + 35.5i)T^{2} \)
53 \( 1 + (-0.335 - 2.33i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (2.84 + 1.83i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (-9.39 - 2.75i)T + (51.3 + 32.9i)T^{2} \)
71 \( 1 + (0.652 + 4.53i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (14.1 + 4.16i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (1.29 - 0.831i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-11.0 + 3.24i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (3.34 - 7.32i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−12.09314423965937143659016321316, −11.37714017725418756255964911138, −10.66305125981359043933021386334, −9.422322079076797164456667239988, −8.101115068421104496825410826033, −6.30518697763214833749202232871, −5.67966877303911123342341726452, −4.00965819901556598862040136387, −3.17362270397126567783807328999, −1.42913304145331320362232038337, 3.75389486944715106292964229195, 4.13610884375106279748721877379, 5.52175046892794645321060007074, 6.56504070787890533843007407336, 7.32555958430795695058271041427, 8.628020226826459157617917522182, 9.624333645161490806229124484450, 11.39625230950406602277792087885, 11.90064601392046943160190688934, 13.30067468438892987709222231972

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