Properties

Label 2-201-201.200-c1-0-3
Degree $2$
Conductor $201$
Sign $-0.694 - 0.719i$
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  + 0.598·2-s + (0.421 + 1.68i)3-s − 1.64·4-s − 3.30·5-s + (0.252 + 1.00i)6-s + 2.20i·7-s − 2.17·8-s + (−2.64 + 1.41i)9-s − 1.97·10-s + 4.09·11-s + (−0.691 − 2.75i)12-s + 5.23i·13-s + 1.31i·14-s + (−1.39 − 5.55i)15-s + 1.97·16-s − 3.25i·17-s + ⋯
L(s)  = 1  + 0.423·2-s + (0.243 + 0.969i)3-s − 0.820·4-s − 1.47·5-s + (0.102 + 0.410i)6-s + 0.832i·7-s − 0.770·8-s + (−0.881 + 0.471i)9-s − 0.625·10-s + 1.23·11-s + (−0.199 − 0.796i)12-s + 1.45i·13-s + 0.352i·14-s + (−0.359 − 1.43i)15-s + 0.494·16-s − 0.788i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.694 - 0.719i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ -0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.326140 + 0.767625i\)
\(L(\frac12)\) \(\approx\) \(0.326140 + 0.767625i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.421 - 1.68i)T \)
67 \( 1 + (-7.09 + 4.07i)T \)
good2 \( 1 - 0.598T + 2T^{2} \)
5 \( 1 + 3.30T + 5T^{2} \)
7 \( 1 - 2.20iT - 7T^{2} \)
11 \( 1 - 4.09T + 11T^{2} \)
13 \( 1 - 5.23iT - 13T^{2} \)
17 \( 1 + 3.25iT - 17T^{2} \)
19 \( 1 + 0.137T + 19T^{2} \)
23 \( 1 + 3.21iT - 23T^{2} \)
29 \( 1 - 9.21iT - 29T^{2} \)
31 \( 1 + 1.43iT - 31T^{2} \)
37 \( 1 - 6.90T + 37T^{2} \)
41 \( 1 + 8.31T + 41T^{2} \)
43 \( 1 - 9.13iT - 43T^{2} \)
47 \( 1 - 12.5iT - 47T^{2} \)
53 \( 1 + 2.45T + 53T^{2} \)
59 \( 1 + 9.02iT - 59T^{2} \)
61 \( 1 + 10.2iT - 61T^{2} \)
71 \( 1 + 0.302iT - 71T^{2} \)
73 \( 1 + 3.46T + 73T^{2} \)
79 \( 1 - 5.02iT - 79T^{2} \)
83 \( 1 + 0.995iT - 83T^{2} \)
89 \( 1 - 5.26iT - 89T^{2} \)
97 \( 1 + 2.53iT - 97T^{2} \)
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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

−12.61748504120343654590533226232, −11.76116265395746323870894526580, −11.22288023379029272456523709018, −9.457162103787411502825126250018, −9.035347322231587466834904553350, −8.112941212039951004280315830345, −6.49850133829910252034814327597, −4.88334432397659033986452633379, −4.22833923211592057788627531062, −3.23840874400094610942362383184, 0.66211561548302443978379683474, 3.45994372741672224645502906490, 4.11734304852623985580232066427, 5.79344221482662669486818183516, 7.13472731379497094184336008784, 8.021408190078592503753449484000, 8.702247755100481293883173349507, 10.17580908666950533742937583159, 11.59424797366205356844912165496, 12.14065192977597770429633386682

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