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Properties

Label 2-201-67.66-c2-0-5
Degree $2$
Conductor $201$
Sign $-0.905 + 0.424i$
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  + 2.74i·2-s + 1.73i·3-s − 3.55·4-s + 0.569i·5-s − 4.76·6-s + 7.88i·7-s + 1.21i·8-s − 2.99·9-s − 1.56·10-s − 7.66i·11-s − 6.16i·12-s + 2.48i·13-s − 21.6·14-s − 0.987·15-s − 17.5·16-s − 20.2·17-s + ⋯
L(s)  = 1  + 1.37i·2-s + 0.577i·3-s − 0.889·4-s + 0.113i·5-s − 0.793·6-s + 1.12i·7-s + 0.152i·8-s − 0.333·9-s − 0.156·10-s − 0.696i·11-s − 0.513i·12-s + 0.191i·13-s − 1.54·14-s − 0.0658·15-s − 1.09·16-s − 1.19·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.905 + 0.424i$
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: \((2,\ 201,\ (\ :1),\ -0.905 + 0.424i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.282177 - 1.26583i\)
\(L(\frac12)\) \(\approx\) \(0.282177 - 1.26583i\)
\(L(2)\) 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 \( 1 + (28.4 + 60.6i)T \)
good2 \( 1 - 2.74iT - 4T^{2} \)
5 \( 1 - 0.569iT - 25T^{2} \)
7 \( 1 - 7.88iT - 49T^{2} \)
11 \( 1 + 7.66iT - 121T^{2} \)
13 \( 1 - 2.48iT - 169T^{2} \)
17 \( 1 + 20.2T + 289T^{2} \)
19 \( 1 - 13.9T + 361T^{2} \)
23 \( 1 - 10.7T + 529T^{2} \)
29 \( 1 + 27.0T + 841T^{2} \)
31 \( 1 - 2.26iT - 961T^{2} \)
37 \( 1 - 53.4T + 1.36e3T^{2} \)
41 \( 1 - 16.0iT - 1.68e3T^{2} \)
43 \( 1 - 44.5iT - 1.84e3T^{2} \)
47 \( 1 + 2.48T + 2.20e3T^{2} \)
53 \( 1 - 86.0iT - 2.80e3T^{2} \)
59 \( 1 + 16.5T + 3.48e3T^{2} \)
61 \( 1 - 97.3iT - 3.72e3T^{2} \)
71 \( 1 - 11.6T + 5.04e3T^{2} \)
73 \( 1 - 58.7T + 5.32e3T^{2} \)
79 \( 1 - 0.870iT - 6.24e3T^{2} \)
83 \( 1 - 47.5T + 6.88e3T^{2} \)
89 \( 1 - 47.1T + 7.92e3T^{2} \)
97 \( 1 - 21.2iT - 9.40e3T^{2} \)
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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

−13.04627260371936153339023650312, −11.63379942841564068438902385913, −10.92974689275094931278122344623, −9.262277738888531827153885957877, −8.820200414738811173104643157994, −7.69235551562060618770200344540, −6.43323516120172338089246159355, −5.64285385844942714626671578016, −4.60883193627550086662589343533, −2.75466911083400343695325812702, 0.74793172230511312905632938547, 2.15295074162342144042999759203, 3.60274670825700897362581839345, 4.78930016499217843686760891123, 6.69130878271539170239714969420, 7.49997241540730643953299732969, 8.987348486714727967610495594647, 9.965216539588767429734761111051, 10.89735337599431889839864304031, 11.53659395844307253119611366631

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