Properties

Label 2-201-201.200-c1-0-1
Degree $2$
Conductor $201$
Sign $0.498 - 0.866i$
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  − 1.18·2-s + (−1.24 − 1.19i)3-s − 0.588·4-s − 2.08·5-s + (1.48 + 1.42i)6-s − 0.284i·7-s + 3.07·8-s + (0.120 + 2.99i)9-s + 2.47·10-s + 1.80·11-s + (0.734 + 0.705i)12-s + 5.82i·13-s + 0.337i·14-s + (2.60 + 2.50i)15-s − 2.47·16-s − 2.11i·17-s + ⋯
L(s)  = 1  − 0.840·2-s + (−0.721 − 0.692i)3-s − 0.294·4-s − 0.932·5-s + (0.605 + 0.582i)6-s − 0.107i·7-s + 1.08·8-s + (0.0401 + 0.999i)9-s + 0.783·10-s + 0.543·11-s + (0.212 + 0.203i)12-s + 1.61i·13-s + 0.0902i·14-s + (0.672 + 0.646i)15-s − 0.619·16-s − 0.513i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.498 - 0.866i$
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: \((2,\ 201,\ (\ :1/2),\ 0.498 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.302396 + 0.174905i\)
\(L(\frac12)\) \(\approx\) \(0.302396 + 0.174905i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.24 + 1.19i)T \)
67 \( 1 + (1.97 + 7.94i)T \)
good2 \( 1 + 1.18T + 2T^{2} \)
5 \( 1 + 2.08T + 5T^{2} \)
7 \( 1 + 0.284iT - 7T^{2} \)
11 \( 1 - 1.80T + 11T^{2} \)
13 \( 1 - 5.82iT - 13T^{2} \)
17 \( 1 + 2.11iT - 17T^{2} \)
19 \( 1 - 3.37T + 19T^{2} \)
23 \( 1 - 5.85iT - 23T^{2} \)
29 \( 1 - 1.18iT - 29T^{2} \)
31 \( 1 - 9.35iT - 31T^{2} \)
37 \( 1 + 9.06T + 37T^{2} \)
41 \( 1 - 6.14T + 41T^{2} \)
43 \( 1 + 1.57iT - 43T^{2} \)
47 \( 1 + 3.59iT - 47T^{2} \)
53 \( 1 + 5.90T + 53T^{2} \)
59 \( 1 - 8.92iT - 59T^{2} \)
61 \( 1 - 3.69iT - 61T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 2.98T + 73T^{2} \)
79 \( 1 - 2.32iT - 79T^{2} \)
83 \( 1 - 10.4iT - 83T^{2} \)
89 \( 1 - 4.96iT - 89T^{2} \)
97 \( 1 - 8.79iT - 97T^{2} \)
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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

−12.22365039279080599469910048120, −11.66544050278195861860998756690, −10.78592422010222835827146637446, −9.522627622918773098604687507565, −8.622006779859164015405432380744, −7.44077642019339214973017341070, −6.92470518484158844103038075935, −5.19372790320873288203904084438, −3.99101094282210696011036077367, −1.40723035470048483597588389237, 0.50186582770773769812744707004, 3.62245062028471141295853358376, 4.67873856187594969173669493120, 5.94610717903470145231197420269, 7.51057696671206252910430877308, 8.358682183023636375056348769906, 9.419953688179502556868856671708, 10.31794723519648703728942397828, 11.05289752552457554605043125517, 12.07749119382157538448457462917

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