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Properties

Label 2-201-201.200-c1-0-13
Degree $2$
Conductor $201$
Sign $-0.272 + 0.962i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.272 + 0.962i$
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: \((2,\ 201,\ (\ :1/2),\ -0.272 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.494993 - 0.654588i\)
\(L(\frac12)\) \(\approx\) \(0.494993 - 0.654588i\)
\(L(\frac{3}{2})\) 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

−12.76637656594276172459368213088, −10.71272678749696592135920941603, −10.36206414914417874633327849659, −9.195781158987246066550739140150, −8.061836236404351352146315942509, −7.22363814719944021384537460357, −5.77139928889454884615969740601, −5.03706055849076096125030232349, −2.69918336779401678554668152290, −0.896311341376370666524894428281, 2.30130994792647939470563859693, 4.28007101224301290110430977301, 5.40661444834268449098774742330, 6.10418353605615717114449454309, 8.122304725124726864867732146557, 9.195882379823604122196634764085, 9.634696614490348033124498448208, 10.40590383975090873433730209555, 11.53601073960471332007588818398, 12.93670219037844643594842513446

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