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Properties

Label 2-201-67.66-c4-0-10
Degree $2$
Conductor $201$
Sign $-0.989 + 0.145i$
Analytic cond. $20.7773$
Root an. cond. $4.55821$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.01i·2-s − 5.19i·3-s − 0.109·4-s + 45.0i·5-s + 20.8·6-s + 18.6i·7-s + 63.7i·8-s − 27·9-s − 180.·10-s + 22.0i·11-s + 0.568i·12-s − 61.5i·13-s − 74.7·14-s + 234.·15-s − 257.·16-s + 70.4·17-s + ⋯
L(s)  = 1  + 1.00i·2-s − 0.577i·3-s − 0.00683·4-s + 1.80i·5-s + 0.579·6-s + 0.380i·7-s + 0.996i·8-s − 0.333·9-s − 1.80·10-s + 0.182i·11-s + 0.00394i·12-s − 0.364i·13-s − 0.381·14-s + 1.04·15-s − 1.00·16-s + 0.243·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.989 + 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.989 + 0.145i$
Analytic conductor: \(20.7773\)
Root analytic conductor: \(4.55821\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :2),\ -0.989 + 0.145i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.621605093\)
\(L(\frac12)\) \(\approx\) \(1.621605093\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19iT \)
67 \( 1 + (652. + 4.44e3i)T \)
good2 \( 1 - 4.01iT - 16T^{2} \)
5 \( 1 - 45.0iT - 625T^{2} \)
7 \( 1 - 18.6iT - 2.40e3T^{2} \)
11 \( 1 - 22.0iT - 1.46e4T^{2} \)
13 \( 1 + 61.5iT - 2.85e4T^{2} \)
17 \( 1 - 70.4T + 8.35e4T^{2} \)
19 \( 1 - 37.5T + 1.30e5T^{2} \)
23 \( 1 - 670.T + 2.79e5T^{2} \)
29 \( 1 + 1.33e3T + 7.07e5T^{2} \)
31 \( 1 + 79.4iT - 9.23e5T^{2} \)
37 \( 1 + 2.03e3T + 1.87e6T^{2} \)
41 \( 1 + 374. iT - 2.82e6T^{2} \)
43 \( 1 - 750. iT - 3.41e6T^{2} \)
47 \( 1 - 2.00e3T + 4.87e6T^{2} \)
53 \( 1 - 3.14e3iT - 7.89e6T^{2} \)
59 \( 1 + 3.37e3T + 1.21e7T^{2} \)
61 \( 1 + 1.76e3iT - 1.38e7T^{2} \)
71 \( 1 + 1.70e3T + 2.54e7T^{2} \)
73 \( 1 - 1.53e3T + 2.83e7T^{2} \)
79 \( 1 - 7.25e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.76e3T + 4.74e7T^{2} \)
89 \( 1 - 9.40e3T + 6.27e7T^{2} \)
97 \( 1 - 1.18e4iT - 8.85e7T^{2} \)
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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

−12.22560859686901491976263383334, −11.19654016135743715766911397347, −10.58219709832116362491879866974, −9.080355416271892494870990151024, −7.69087052865260983059480355946, −7.19532872897460714248842282136, −6.33790520145321742867250795619, −5.44965914342383268291655684914, −3.24163210768572514954604332150, −2.17063337152430645818886377669, 0.55148020662648959694093681281, 1.72069721787135116627460583034, 3.51191767496045257726610565970, 4.51574737404425132479625846314, 5.58266070825731697935326460810, 7.28576988612231553060789927092, 8.748900953024521937357689575418, 9.341401249006577387733856648343, 10.35189291926502947138355700547, 11.34152555711247214897796122942

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