↑

Properties

Label 2-201-201.164-c1-0-17
Degree $2$
Conductor $201$
Sign $-0.0505 + 0.998i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.144 + 0.249i)2-s + (0.169 − 1.72i)3-s + (0.958 − 1.66i)4-s − 2.85·5-s + (0.454 − 0.205i)6-s + (0.946 + 0.546i)7-s + 1.12·8-s + (−2.94 − 0.583i)9-s + (−0.411 − 0.713i)10-s + (2.36 − 4.09i)11-s + (−2.69 − 1.93i)12-s + (0.0274 − 0.0158i)13-s + 0.314i·14-s + (−0.484 + 4.92i)15-s + (−1.75 − 3.03i)16-s + (−4.02 + 2.32i)17-s + ⋯
L(s)  = 1  + (0.101 + 0.176i)2-s + (0.0977 − 0.995i)3-s + (0.479 − 0.830i)4-s − 1.27·5-s + (0.185 − 0.0840i)6-s + (0.357 + 0.206i)7-s + 0.398·8-s + (−0.980 − 0.194i)9-s + (−0.130 − 0.225i)10-s + (0.712 − 1.23i)11-s + (−0.779 − 0.558i)12-s + (0.00761 − 0.00439i)13-s + 0.0841i·14-s + (−0.125 + 1.27i)15-s + (−0.438 − 0.759i)16-s + (−0.975 + 0.563i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0505 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0505 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.0505 + 0.998i$
Analytic conductor: \(1.60499\)
Root analytic conductor: \(1.26688\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1/2),\ -0.0505 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.815185 - 0.857459i\)
\(L(\frac12)\) \(\approx\) \(0.815185 - 0.857459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.169 + 1.72i)T \)
67 \( 1 + (-5.74 - 5.83i)T \)
good2 \( 1 + (-0.144 - 0.249i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2.85T + 5T^{2} \)
7 \( 1 + (-0.946 - 0.546i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.36 + 4.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0274 + 0.0158i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.02 - 2.32i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.793 - 1.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.26 + 3.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.37 - 4.83i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.36 - 3.67i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.64 - 4.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.791 - 1.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 2.03iT - 43T^{2} \)
47 \( 1 + (0.709 + 0.409i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.06T + 53T^{2} \)
59 \( 1 - 3.79iT - 59T^{2} \)
61 \( 1 + (-6.20 + 3.58i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (12.0 + 6.97i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.673 + 1.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (14.3 + 8.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.55 + 3.78i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.67iT - 89T^{2} \)
97 \( 1 + (6.46 - 3.73i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.94881112295598919469957420321, −11.44372672287009463040162473077, −10.63333073892844173860317348698, −8.769285736850384466468424084612, −8.182142561985375237942973874783, −6.88026074928320426914815121915, −6.27111607439631097755229200320, −4.77617834235861056768273362994, −3.02764196633080422887612893280, −1.10028026212530379527183644077, 2.78988533914750607284465479198, 4.13237600742162963030063133508, 4.61714221638841429322691588995, 6.80578042739175375341151012259, 7.71190644539784395238136891038, 8.667931084845778140887174169613, 9.782133122790233381067227588235, 11.17486032599658584628813137406, 11.50223597714187293356552087393, 12.36669438876739781730262043705

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