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Properties

Label 2-201-201.38-c1-0-13
Degree $2$
Conductor $201$
Sign $0.862 + 0.505i$
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  − 1.73i·3-s + (1 + 1.73i)4-s + (3 − 1.73i)7-s − 2.99·9-s + (2.99 − 1.73i)12-s + (1.5 + 0.866i)13-s + (−1.99 + 3.46i)16-s + (4 − 6.92i)19-s + (−2.99 − 5.19i)21-s − 5·25-s + 5.19i·27-s + (6 + 3.46i)28-s + (−7.5 + 4.33i)31-s + (−2.99 − 5.19i)36-s + (−5 + 8.66i)37-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.5 + 0.866i)4-s + (1.13 − 0.654i)7-s − 0.999·9-s + (0.866 − 0.499i)12-s + (0.416 + 0.240i)13-s + (−0.499 + 0.866i)16-s + (0.917 − 1.58i)19-s + (−0.654 − 1.13i)21-s − 25-s + 0.999i·27-s + (1.13 + 0.654i)28-s + (−1.34 + 0.777i)31-s + (−0.499 − 0.866i)36-s + (−0.821 + 1.42i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.862 + 0.505i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.862 + 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34812 - 0.365778i\)
\(L(\frac12)\) \(\approx\) \(1.34812 - 0.365778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
67 \( 1 + (2.5 + 7.79i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + (-3 + 1.73i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4 + 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (7.5 + 4.33i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.5 + 14.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (4.5 + 2.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.28451661759237724100706729835, −11.42294118837224217405332456831, −10.96024311425129136224749613788, −9.079871547189229652543877367797, −8.015363630144611141502948401274, −7.41564914703290326250863712393, −6.47848823344714132384351381309, −4.86650631054172030737067667953, −3.23236307143343293489171197126, −1.66887642201510692538169405140, 1.99469443071340221084861994500, 3.83491271730638232846916347580, 5.41922484150236154904235423533, 5.71926691402001289613629893767, 7.58034432943335284601240501042, 8.718983411824292301617037917654, 9.727610855142279157051835499105, 10.61447750941184741502720916916, 11.38189293323274019619792288494, 12.12963089637884718109286515957

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