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Properties

Label 2-201-67.66-c4-0-39
Degree $2$
Conductor $201$
Sign $-0.214 + 0.976i$
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  + 3.33i·2-s − 5.19i·3-s + 4.86·4-s − 44.8i·5-s + 17.3·6-s − 17.0i·7-s + 69.6i·8-s − 27·9-s + 149.·10-s − 146. i·11-s − 25.2i·12-s + 86.6i·13-s + 56.7·14-s − 233.·15-s − 154.·16-s + 506.·17-s + ⋯
L(s)  = 1  + 0.834i·2-s − 0.577i·3-s + 0.303·4-s − 1.79i·5-s + 0.481·6-s − 0.346i·7-s + 1.08i·8-s − 0.333·9-s + 1.49·10-s − 1.21i·11-s − 0.175i·12-s + 0.512i·13-s + 0.289·14-s − 1.03·15-s − 0.603·16-s + 1.75·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.214 + 0.976i$
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: \((2,\ 201,\ (\ :2),\ -0.214 + 0.976i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.637082455\)
\(L(\frac12)\) \(\approx\) \(1.637082455\)
\(L(3)\) 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 \( 1 + (4.38e3 + 965. i)T \)
good2 \( 1 - 3.33iT - 16T^{2} \)
5 \( 1 + 44.8iT - 625T^{2} \)
7 \( 1 + 17.0iT - 2.40e3T^{2} \)
11 \( 1 + 146. iT - 1.46e4T^{2} \)
13 \( 1 - 86.6iT - 2.85e4T^{2} \)
17 \( 1 - 506.T + 8.35e4T^{2} \)
19 \( 1 + 586.T + 1.30e5T^{2} \)
23 \( 1 + 838.T + 2.79e5T^{2} \)
29 \( 1 + 119.T + 7.07e5T^{2} \)
31 \( 1 + 1.03e3iT - 9.23e5T^{2} \)
37 \( 1 + 286.T + 1.87e6T^{2} \)
41 \( 1 + 840. iT - 2.82e6T^{2} \)
43 \( 1 + 2.63e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.63e3T + 4.87e6T^{2} \)
53 \( 1 - 3.75e3iT - 7.89e6T^{2} \)
59 \( 1 + 533.T + 1.21e7T^{2} \)
61 \( 1 + 1.94e3iT - 1.38e7T^{2} \)
71 \( 1 - 761.T + 2.54e7T^{2} \)
73 \( 1 - 4.63e3T + 2.83e7T^{2} \)
79 \( 1 + 8.84e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.67e3T + 4.74e7T^{2} \)
89 \( 1 + 1.18e4T + 6.27e7T^{2} \)
97 \( 1 - 4.55e3iT - 8.85e7T^{2} \)
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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

−11.89875670556666142251186245405, −10.56455610359896840816914207235, −9.029523384300568726118038641645, −8.263111854981115223631307534039, −7.61033650391716985945522377453, −6.08722128754203738268681786868, −5.55156000424299137044917187655, −4.06336508665155712066043818423, −1.90918475967583345524760627114, −0.54067709889245896014475766674, 2.05478386779860414585492840574, 3.00421576677343079459500581082, 4.00965216223021916137442694631, 5.90062817042034145846733768330, 6.88461404437399426068063458712, 7.920974179307737591708865886632, 9.802971089900849239844161746234, 10.22824797211827289920441354767, 10.86951951814046568076329350963, 11.92450051634908275202255599598

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