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Properties

Label 2-201-67.66-c2-0-12
Degree $2$
Conductor $201$
Sign $0.698 + 0.715i$
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  − 0.465i·2-s + 1.73i·3-s + 3.78·4-s − 1.28i·5-s + 0.806·6-s − 11.7i·7-s − 3.62i·8-s − 2.99·9-s − 0.600·10-s − 17.7i·11-s + 6.55i·12-s + 20.8i·13-s − 5.47·14-s + 2.23·15-s + 13.4·16-s + 17.2·17-s + ⋯
L(s)  = 1  − 0.232i·2-s + 0.577i·3-s + 0.945·4-s − 0.257i·5-s + 0.134·6-s − 1.67i·7-s − 0.453i·8-s − 0.333·9-s − 0.0600·10-s − 1.60i·11-s + 0.546i·12-s + 1.60i·13-s − 0.390·14-s + 0.148·15-s + 0.840·16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.698 + 0.715i$
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: \((2,\ 201,\ (\ :1),\ 0.698 + 0.715i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.70720 - 0.719365i\)
\(L(\frac12)\) \(\approx\) \(1.70720 - 0.719365i\)
\(L(2)\) 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 \( 1 + (47.9 - 46.7i)T \)
good2 \( 1 + 0.465iT - 4T^{2} \)
5 \( 1 + 1.28iT - 25T^{2} \)
7 \( 1 + 11.7iT - 49T^{2} \)
11 \( 1 + 17.7iT - 121T^{2} \)
13 \( 1 - 20.8iT - 169T^{2} \)
17 \( 1 - 17.2T + 289T^{2} \)
19 \( 1 - 5.65T + 361T^{2} \)
23 \( 1 + 14.7T + 529T^{2} \)
29 \( 1 + 51.4T + 841T^{2} \)
31 \( 1 - 18.2iT - 961T^{2} \)
37 \( 1 - 54.4T + 1.36e3T^{2} \)
41 \( 1 - 64.3iT - 1.68e3T^{2} \)
43 \( 1 + 21.7iT - 1.84e3T^{2} \)
47 \( 1 - 12.1T + 2.20e3T^{2} \)
53 \( 1 + 1.46iT - 2.80e3T^{2} \)
59 \( 1 + 7.90T + 3.48e3T^{2} \)
61 \( 1 - 25.0iT - 3.72e3T^{2} \)
71 \( 1 + 43.8T + 5.04e3T^{2} \)
73 \( 1 + 62.2T + 5.32e3T^{2} \)
79 \( 1 - 83.6iT - 6.24e3T^{2} \)
83 \( 1 - 106.T + 6.88e3T^{2} \)
89 \( 1 + 88.0T + 7.92e3T^{2} \)
97 \( 1 + 121. iT - 9.40e3T^{2} \)
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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

−11.68766313040870242813924451784, −11.14025265921083300250247068877, −10.34216073945363112442080875611, −9.345866621185467975767392961944, −7.968882788715981396244679849738, −6.96813187166933956495589518894, −5.86007925855066107261864212134, −4.20745213642713128239660363740, −3.27304131263182891816698327672, −1.16867236710222298571270530843, 1.98986886201029718662168582628, 2.96480276491526066286162340843, 5.39084851649772899742859086939, 6.01873481465678810368337501413, 7.38072374679618773573355518761, 7.947259365705329691038904405228, 9.364645175204109352471120901050, 10.44961556192352132867302311722, 11.63887949525507632571814993468, 12.37585795533941902197510079843

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