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Properties

Label 2-201-67.37-c3-0-27
Degree $2$
Conductor $201$
Sign $-0.971 - 0.237i$
Analytic cond. $11.8593$
Root an. cond. $3.44374$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.07 − 3.59i)2-s + 3·3-s + (−4.59 + 7.95i)4-s + 0.952·5-s + (−6.21 − 10.7i)6-s + (4.18 − 7.25i)7-s + 4.93·8-s + 9·9-s + (−1.97 − 3.42i)10-s + (9.85 − 17.0i)11-s + (−13.7 + 23.8i)12-s + (−7.29 − 12.6i)13-s − 34.7·14-s + 2.85·15-s + (26.5 + 45.9i)16-s + (−44.5 − 77.1i)17-s + ⋯
L(s)  = 1  + (−0.732 − 1.26i)2-s + 0.577·3-s + (−0.574 + 0.994i)4-s + 0.0852·5-s + (−0.423 − 0.732i)6-s + (0.226 − 0.391i)7-s + 0.218·8-s + 0.333·9-s + (−0.0624 − 0.108i)10-s + (0.270 − 0.467i)11-s + (−0.331 + 0.574i)12-s + (−0.155 − 0.269i)13-s − 0.662·14-s + 0.0491·15-s + (0.414 + 0.717i)16-s + (−0.635 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.237i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.971 - 0.237i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.971 - 0.237i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.118905 + 0.989087i\)
\(L(\frac12)\) \(\approx\) \(0.118905 + 0.989087i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
67 \( 1 + (-355. - 417. i)T \)
good2 \( 1 + (2.07 + 3.59i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 0.952T + 125T^{2} \)
7 \( 1 + (-4.18 + 7.25i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-9.85 + 17.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (7.29 + 12.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (44.5 + 77.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (18.6 + 32.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (83.9 + 145. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-18.5 + 32.0i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (34.4 - 59.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-132. - 229. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-30.5 + 52.8i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + 312.T + 7.95e4T^{2} \)
47 \( 1 + (3.45 - 5.97i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 305.T + 1.48e5T^{2} \)
59 \( 1 - 166.T + 2.05e5T^{2} \)
61 \( 1 + (177. + 306. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
71 \( 1 + (-401. + 695. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (335. + 580. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-456. + 791. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (177. + 307. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.13e3T + 7.04e5T^{2} \)
97 \( 1 + (157. + 271. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.33992498138388364655600695796, −10.44729622042916582043572440230, −9.609490211689051777750261234140, −8.745319499586537190822662774488, −7.84733793372081066625265683884, −6.39730701170665200048458994251, −4.51496037878312518181921674968, −3.17961524358198997194847722314, −2.05434405222247447658396235647, −0.49428392314416051553527162049, 1.93949585550132541846215450686, 3.92723208498762619564739385622, 5.54576687547355157343993417156, 6.54726726132721027823392747715, 7.63680625732961916768558103345, 8.356738349667383625514652581184, 9.293659005188065060219000701727, 10.02201473938796748091658754625, 11.53177248360869412402070977265, 12.63204433684355062035021828921

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