Properties

Label 2-201-67.64-c1-0-8
Degree $2$
Conductor $201$
Sign $0.991 - 0.133i$
Analytic cond. $1.60499$
Root an. cond. $1.26688$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.15 + 0.632i)2-s + (0.142 − 0.989i)3-s + (2.55 + 1.64i)4-s + (−0.146 − 0.319i)5-s + (0.932 − 2.04i)6-s + (−0.489 − 0.143i)7-s + (1.52 + 1.76i)8-s + (−0.959 − 0.281i)9-s + (−0.112 − 0.781i)10-s + (0.961 + 2.10i)11-s + (1.99 − 2.29i)12-s + (−1.11 + 1.28i)13-s + (−0.964 − 0.619i)14-s + (−0.337 + 0.0990i)15-s + (−0.348 − 0.763i)16-s + (−3.94 + 2.53i)17-s + ⋯
L(s)  = 1  + (1.52 + 0.447i)2-s + (0.0821 − 0.571i)3-s + (1.27 + 0.821i)4-s + (−0.0653 − 0.143i)5-s + (0.380 − 0.833i)6-s + (−0.185 − 0.0543i)7-s + (0.540 + 0.623i)8-s + (−0.319 − 0.0939i)9-s + (−0.0355 − 0.247i)10-s + (0.289 + 0.634i)11-s + (0.574 − 0.663i)12-s + (−0.309 + 0.357i)13-s + (−0.257 − 0.165i)14-s + (−0.0871 + 0.0255i)15-s + (−0.0872 − 0.190i)16-s + (−0.956 + 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.991 - 0.133i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ 0.991 - 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43233 + 0.163098i\)
\(L(\frac12)\) \(\approx\) \(2.43233 + 0.163098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (-8.16 + 0.560i)T \)
good2 \( 1 + (-2.15 - 0.632i)T + (1.68 + 1.08i)T^{2} \)
5 \( 1 + (0.146 + 0.319i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (0.489 + 0.143i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (-0.961 - 2.10i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (1.11 - 1.28i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (3.94 - 2.53i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (2.21 - 0.651i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (-0.229 + 1.59i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + 2.16T + 29T^{2} \)
31 \( 1 + (-1.40 - 1.62i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 - 5.02T + 37T^{2} \)
41 \( 1 + (3.00 - 1.92i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-5.14 + 3.30i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + (0.984 - 6.84i)T + (-45.0 - 13.2i)T^{2} \)
53 \( 1 + (0.767 + 0.493i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-3.80 - 4.38i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (-3.68 + 8.06i)T + (-39.9 - 46.1i)T^{2} \)
71 \( 1 + (-7.41 - 4.76i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-6.17 + 13.5i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (0.963 - 1.11i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (1.73 + 3.80i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-1.97 - 13.7i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + 5.56T + 97T^{2} \)
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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

−12.68310155110996931069210766148, −12.04730559609790341975040512867, −10.92334839194692423695797860697, −9.423659700187687481094855389837, −8.119078654297551339110505994203, −6.84663888393437562152620937464, −6.33070252182462706904511806556, −4.91694772098090392672248154346, −3.95921136369444638555479992168, −2.36987034381965948139123368446, 2.58882953805122234302431307310, 3.67576006849097792132415020431, 4.75227698074216987659647061661, 5.76243498973167149822851529751, 6.89043361091053579567435029753, 8.551711253927755841186877445878, 9.693358439711783898793545449956, 11.02984475340431306799019346371, 11.38569941320319984236249661559, 12.61953731819299018491116607637

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