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Properties

Label 2-201-67.66-c2-0-2
Degree $2$
Conductor $201$
Sign $-0.925 + 0.378i$
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  + 3.12i·2-s − 1.73i·3-s − 5.75·4-s + 8.39i·5-s + 5.41·6-s − 3.67i·7-s − 5.48i·8-s − 2.99·9-s − 26.2·10-s + 15.4i·11-s + 9.97i·12-s − 12.0i·13-s + 11.4·14-s + 14.5·15-s − 5.88·16-s − 27.4·17-s + ⋯
L(s)  = 1  + 1.56i·2-s − 0.577i·3-s − 1.43·4-s + 1.67i·5-s + 0.901·6-s − 0.524i·7-s − 0.685i·8-s − 0.333·9-s − 2.62·10-s + 1.40i·11-s + 0.830i·12-s − 0.924i·13-s + 0.819·14-s + 0.969·15-s − 0.367·16-s − 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.925 + 0.378i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ -0.925 + 0.378i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.208008 - 1.05828i\)
\(L(\frac12)\) \(\approx\) \(0.208008 - 1.05828i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
67 \( 1 + (-25.3 - 62.0i)T \)
good2 \( 1 - 3.12iT - 4T^{2} \)
5 \( 1 - 8.39iT - 25T^{2} \)
7 \( 1 + 3.67iT - 49T^{2} \)
11 \( 1 - 15.4iT - 121T^{2} \)
13 \( 1 + 12.0iT - 169T^{2} \)
17 \( 1 + 27.4T + 289T^{2} \)
19 \( 1 - 19.9T + 361T^{2} \)
23 \( 1 + 32.3T + 529T^{2} \)
29 \( 1 - 40.5T + 841T^{2} \)
31 \( 1 - 33.8iT - 961T^{2} \)
37 \( 1 - 47.6T + 1.36e3T^{2} \)
41 \( 1 - 29.9iT - 1.68e3T^{2} \)
43 \( 1 - 39.0iT - 1.84e3T^{2} \)
47 \( 1 - 32.9T + 2.20e3T^{2} \)
53 \( 1 + 18.7iT - 2.80e3T^{2} \)
59 \( 1 - 6.08T + 3.48e3T^{2} \)
61 \( 1 - 76.6iT - 3.72e3T^{2} \)
71 \( 1 + 108.T + 5.04e3T^{2} \)
73 \( 1 - 71.7T + 5.32e3T^{2} \)
79 \( 1 - 5.18iT - 6.24e3T^{2} \)
83 \( 1 - 34.3T + 6.88e3T^{2} \)
89 \( 1 + 1.95T + 7.92e3T^{2} \)
97 \( 1 + 24.1iT - 9.40e3T^{2} \)
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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

−13.22641394319255126251432763242, −11.83671622315088420088289789376, −10.67024853156660636896124994050, −9.757101801284648243473455841161, −8.187890412515177048078512339023, −7.31649041137385659386409256910, −6.85584548800339807814589817119, −5.98207194713887152411408006900, −4.44135594116613891597150544474, −2.60044133101155019495856166008, 0.60931141857897172987765503550, 2.24733090816457447512724994112, 3.89746214389665565765003306931, 4.70902920819221105912309879017, 5.96909586982040095854043404456, 8.370613814547067070906339577664, 9.025536450067257487746648601609, 9.620730744190893672273817842055, 10.92174922782682869123742561921, 11.75157524928419036714742942902

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