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Properties

Label 2-201-67.66-c4-0-44
Degree $2$
Conductor $201$
Sign $-0.839 - 0.543i$
Analytic cond. $20.7773$
Root an. cond. $4.55821$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.88i·2-s − 5.19i·3-s + 0.868·4-s − 32.6i·5-s − 20.2·6-s − 59.7i·7-s − 65.6i·8-s − 27·9-s − 126.·10-s + 112. i·11-s − 4.51i·12-s − 160. i·13-s − 232.·14-s − 169.·15-s − 241.·16-s + 547.·17-s + ⋯
L(s)  = 1  − 0.972i·2-s − 0.577i·3-s + 0.0542·4-s − 1.30i·5-s − 0.561·6-s − 1.22i·7-s − 1.02i·8-s − 0.333·9-s − 1.26·10-s + 0.927i·11-s − 0.0313i·12-s − 0.951i·13-s − 1.18·14-s − 0.753·15-s − 0.942·16-s + 1.89·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(20.7773\)
Root analytic conductor: \(4.55821\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :2),\ -0.839 - 0.543i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.100101986\)
\(L(\frac12)\) \(\approx\) \(2.100101986\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19iT \)
67 \( 1 + (-2.43e3 + 3.76e3i)T \)
good2 \( 1 + 3.88iT - 16T^{2} \)
5 \( 1 + 32.6iT - 625T^{2} \)
7 \( 1 + 59.7iT - 2.40e3T^{2} \)
11 \( 1 - 112. iT - 1.46e4T^{2} \)
13 \( 1 + 160. iT - 2.85e4T^{2} \)
17 \( 1 - 547.T + 8.35e4T^{2} \)
19 \( 1 - 521.T + 1.30e5T^{2} \)
23 \( 1 + 830.T + 2.79e5T^{2} \)
29 \( 1 + 386.T + 7.07e5T^{2} \)
31 \( 1 - 310. iT - 9.23e5T^{2} \)
37 \( 1 - 2.25e3T + 1.87e6T^{2} \)
41 \( 1 - 3.26e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.35e3iT - 3.41e6T^{2} \)
47 \( 1 + 198.T + 4.87e6T^{2} \)
53 \( 1 - 1.17e3iT - 7.89e6T^{2} \)
59 \( 1 - 3.54e3T + 1.21e7T^{2} \)
61 \( 1 - 1.59e3iT - 1.38e7T^{2} \)
71 \( 1 + 9.01e3T + 2.54e7T^{2} \)
73 \( 1 + 2.76e3T + 2.83e7T^{2} \)
79 \( 1 - 1.00e3iT - 3.89e7T^{2} \)
83 \( 1 + 2.09e3T + 4.74e7T^{2} \)
89 \( 1 - 4.11e3T + 6.27e7T^{2} \)
97 \( 1 + 7.65e3iT - 8.85e7T^{2} \)
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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

−11.45787409303033976483197225210, −10.02511581174979881301072857065, −9.761613532230801879277101286343, −7.926063685027810605234723720611, −7.43349894288194044223805629891, −5.80288726439581289067349100428, −4.44211326097815657854536448248, −3.17879466372573372263102690776, −1.39491449698722504270122817400, −0.813939009073740021818381907062, 2.39907385256347151390382633772, 3.51382617460840859172535648119, 5.58437069446484381581685250950, 5.93159446981571299393340116108, 7.22460938388691935306596635676, 8.141880688845389596142209200824, 9.312592296290864625226457862295, 10.34405053993781617890626852436, 11.57363851955316714429105814777, 11.85676515278203671335991142831

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