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Properties

Label 2-201-201.62-c0-0-0
Degree $2$
Conductor $201$
Sign $0.977 - 0.209i$
Analytic cond. $0.100312$
Root an. cond. $0.316720$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.415 + 0.909i)3-s + (−0.142 − 0.989i)4-s + (0.186 − 0.215i)7-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)12-s + (−1.10 + 0.708i)13-s + (−0.959 + 0.281i)16-s + (−0.544 − 0.627i)19-s + (0.273 + 0.0801i)21-s + (0.841 − 0.540i)25-s + (−0.959 − 0.281i)27-s + (−0.239 − 0.153i)28-s + (−1.61 − 1.03i)31-s + (0.841 + 0.540i)36-s + 1.68·37-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)3-s + (−0.142 − 0.989i)4-s + (0.186 − 0.215i)7-s + (−0.654 + 0.755i)9-s + (0.841 − 0.540i)12-s + (−1.10 + 0.708i)13-s + (−0.959 + 0.281i)16-s + (−0.544 − 0.627i)19-s + (0.273 + 0.0801i)21-s + (0.841 − 0.540i)25-s + (−0.959 − 0.281i)27-s + (−0.239 − 0.153i)28-s + (−1.61 − 1.03i)31-s + (0.841 + 0.540i)36-s + 1.68·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.977 - 0.209i$
Analytic conductor: \(0.100312\)
Root analytic conductor: \(0.316720\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :0),\ 0.977 - 0.209i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7346934610\)
\(L(\frac12)\) \(\approx\) \(0.7346934610\)
\(L(1)\) 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 \( 1 + (-0.841 + 0.540i)T \)
good2 \( 1 + (0.142 + 0.989i)T^{2} \)
5 \( 1 + (-0.841 + 0.540i)T^{2} \)
7 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
11 \( 1 + (-0.841 + 0.540i)T^{2} \)
13 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
23 \( 1 + (0.654 - 0.755i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
37 \( 1 - 1.68T + T^{2} \)
41 \( 1 + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.654 - 0.755i)T^{2} \)
53 \( 1 + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.959 - 0.281i)T^{2} \)
73 \( 1 + (-1.84 - 0.540i)T + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (-1.41 + 0.909i)T + (0.415 - 0.909i)T^{2} \)
83 \( 1 + (-0.841 + 0.540i)T^{2} \)
89 \( 1 + (0.654 + 0.755i)T^{2} \)
97 \( 1 - 0.830T + T^{2} \)
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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

−12.87437398212227162990822184890, −11.34197574747998225003595613383, −10.73657278335023600728076315970, −9.618862533437738950827584256688, −9.188811016894397750878454323383, −7.79231444043136811084452426080, −6.38800713140840286698874402626, −5.04279795599805844043680312393, −4.29029707440931201342438965251, −2.38980598041041203657329163154, 2.35399982463227788155007059926, 3.58709128776213239155855266998, 5.27707622151570124240862432831, 6.82981130205455755479149753733, 7.65174021993838534597512107313, 8.461067397530067806453715320392, 9.397307942538483648845155435368, 10.90525113928465373630043900151, 12.12878107979351982848278764513, 12.57189597796151782298207190947

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