Properties

Label 2-201-3.2-c2-0-33
Degree $2$
Conductor $201$
Sign $-0.972 + 0.233i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.37i·2-s + (−2.91 + 0.699i)3-s − 1.64·4-s − 3.76i·5-s + (1.66 + 6.93i)6-s + 9.72·7-s − 5.58i·8-s + (8.02 − 4.08i)9-s − 8.95·10-s − 6.42i·11-s + (4.80 − 1.15i)12-s − 19.9·13-s − 23.1i·14-s + (2.63 + 10.9i)15-s − 19.8·16-s + 12.2i·17-s + ⋯
L(s)  = 1  − 1.18i·2-s + (−0.972 + 0.233i)3-s − 0.412·4-s − 0.753i·5-s + (0.277 + 1.15i)6-s + 1.38·7-s − 0.698i·8-s + (0.891 − 0.453i)9-s − 0.895·10-s − 0.584i·11-s + (0.400 − 0.0961i)12-s − 1.53·13-s − 1.65i·14-s + (0.175 + 0.732i)15-s − 1.24·16-s + 0.718i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.972 + 0.233i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.972 + 0.233i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.131448 - 1.11153i\)
\(L(\frac12)\) \(\approx\) \(0.131448 - 1.11153i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.91 - 0.699i)T \)
67 \( 1 - 8.18T \)
good2 \( 1 + 2.37iT - 4T^{2} \)
5 \( 1 + 3.76iT - 25T^{2} \)
7 \( 1 - 9.72T + 49T^{2} \)
11 \( 1 + 6.42iT - 121T^{2} \)
13 \( 1 + 19.9T + 169T^{2} \)
17 \( 1 - 12.2iT - 289T^{2} \)
19 \( 1 + 16.2T + 361T^{2} \)
23 \( 1 + 29.8iT - 529T^{2} \)
29 \( 1 + 38.6iT - 841T^{2} \)
31 \( 1 - 11.7T + 961T^{2} \)
37 \( 1 - 14.6T + 1.36e3T^{2} \)
41 \( 1 + 6.17iT - 1.68e3T^{2} \)
43 \( 1 + 11.1T + 1.84e3T^{2} \)
47 \( 1 + 1.34iT - 2.20e3T^{2} \)
53 \( 1 - 74.6iT - 2.80e3T^{2} \)
59 \( 1 - 58.8iT - 3.48e3T^{2} \)
61 \( 1 - 36.2T + 3.72e3T^{2} \)
71 \( 1 + 128. iT - 5.04e3T^{2} \)
73 \( 1 - 105.T + 5.32e3T^{2} \)
79 \( 1 - 117.T + 6.24e3T^{2} \)
83 \( 1 + 56.8iT - 6.88e3T^{2} \)
89 \( 1 + 71.5iT - 7.92e3T^{2} \)
97 \( 1 - 118.T + 9.40e3T^{2} \)
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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

−11.83316297869399198831316034813, −10.84487824407582219536286882992, −10.29694309782367579580576649211, −9.073440539438911023002680115401, −7.84967006158597810101528936165, −6.35864188764146663282172775581, −4.90681153578782208601717949089, −4.28023312215724257629358688623, −2.16885704166747345578701868799, −0.72093869172424618446988044954, 2.10687321462990366764024242198, 4.78937533058225514108238073104, 5.29813398761494378659409152407, 6.78143543509393362495527805102, 7.24774278697650721746933576152, 8.139106298363207717251697829886, 9.773191313222462900184914326205, 10.98122404393897767535524773317, 11.53016163845546945367870713140, 12.55908472289033559615642851145

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