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Properties

Label 2-201-3.2-c2-0-4
Degree $2$
Conductor $201$
Sign $-0.854 + 0.519i$
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  + 3.48i·2-s + (2.56 − 1.55i)3-s − 8.17·4-s + 4.65i·5-s + (5.43 + 8.94i)6-s − 13.4·7-s − 14.5i·8-s + (4.14 − 7.99i)9-s − 16.2·10-s + 16.1i·11-s + (−20.9 + 12.7i)12-s + 3.95·13-s − 46.9i·14-s + (7.26 + 11.9i)15-s + 18.0·16-s + 4.42i·17-s + ⋯
L(s)  = 1  + 1.74i·2-s + (0.854 − 0.519i)3-s − 2.04·4-s + 0.931i·5-s + (0.906 + 1.49i)6-s − 1.92·7-s − 1.81i·8-s + (0.460 − 0.887i)9-s − 1.62·10-s + 1.47i·11-s + (−1.74 + 1.06i)12-s + 0.303·13-s − 3.35i·14-s + (0.484 + 0.796i)15-s + 1.12·16-s + 0.260i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.854 + 0.519i$
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: \((2,\ 201,\ (\ :1),\ -0.854 + 0.519i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.296143 - 1.05715i\)
\(L(\frac12)\) \(\approx\) \(0.296143 - 1.05715i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.56 + 1.55i)T \)
67 \( 1 + 8.18T \)
good2 \( 1 - 3.48iT - 4T^{2} \)
5 \( 1 - 4.65iT - 25T^{2} \)
7 \( 1 + 13.4T + 49T^{2} \)
11 \( 1 - 16.1iT - 121T^{2} \)
13 \( 1 - 3.95T + 169T^{2} \)
17 \( 1 - 4.42iT - 289T^{2} \)
19 \( 1 + 22.7T + 361T^{2} \)
23 \( 1 + 0.463iT - 529T^{2} \)
29 \( 1 - 49.1iT - 841T^{2} \)
31 \( 1 - 51.3T + 961T^{2} \)
37 \( 1 - 27.0T + 1.36e3T^{2} \)
41 \( 1 + 13.8iT - 1.68e3T^{2} \)
43 \( 1 + 11.4T + 1.84e3T^{2} \)
47 \( 1 - 65.3iT - 2.20e3T^{2} \)
53 \( 1 + 10.2iT - 2.80e3T^{2} \)
59 \( 1 - 1.70iT - 3.48e3T^{2} \)
61 \( 1 + 29.9T + 3.72e3T^{2} \)
71 \( 1 - 80.0iT - 5.04e3T^{2} \)
73 \( 1 + 21.0T + 5.32e3T^{2} \)
79 \( 1 + 89.2T + 6.24e3T^{2} \)
83 \( 1 - 51.8iT - 6.88e3T^{2} \)
89 \( 1 - 0.0561iT - 7.92e3T^{2} \)
97 \( 1 - 46.4T + 9.40e3T^{2} \)
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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

−12.98304529467282359648797962421, −12.56174113771716897945139911987, −10.28120321437354819186260327175, −9.498471478985833306825957041978, −8.580323456906322067263914099771, −7.32386535518351815674266808817, −6.73319857811622171998259046090, −6.22847469104644104483715993080, −4.23036378808100325154537548450, −2.86956042801545085497522167218, 0.56068648317806226602328517098, 2.62247410079785317317669883645, 3.49379454272436691456964902550, 4.45333496861746572552426099235, 6.15624444696594697750492974043, 8.393346391178549297147721036644, 8.978305528642170638691343982547, 9.810192484549157449457688893595, 10.46812943333341142314924658266, 11.65935193544417310270100930661

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