Properties

Label 2-201-67.29-c3-0-10
Degree $2$
Conductor $201$
Sign $-0.117 - 0.993i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0760 − 0.131i)2-s − 3·3-s + (3.98 + 6.90i)4-s + 2.82·5-s + (−0.228 + 0.395i)6-s + (−1.28 − 2.22i)7-s + 2.42·8-s + 9·9-s + (0.215 − 0.372i)10-s + (16.0 + 27.7i)11-s + (−11.9 − 20.7i)12-s + (2.60 − 4.50i)13-s − 0.390·14-s − 8.48·15-s + (−31.7 + 54.9i)16-s + (−4.88 + 8.46i)17-s + ⋯
L(s)  = 1  + (0.0268 − 0.0465i)2-s − 0.577·3-s + (0.498 + 0.863i)4-s + 0.252·5-s + (−0.0155 + 0.0268i)6-s + (−0.0692 − 0.120i)7-s + 0.107·8-s + 0.333·9-s + (0.00680 − 0.0117i)10-s + (0.439 + 0.760i)11-s + (−0.287 − 0.498i)12-s + (0.0555 − 0.0961i)13-s − 0.00745·14-s − 0.146·15-s + (−0.495 + 0.858i)16-s + (−0.0697 + 0.120i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.117 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $-0.117 - 0.993i$
Analytic conductor: \(11.8593\)
Root analytic conductor: \(3.44374\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :3/2),\ -0.117 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.975286 + 1.09794i\)
\(L(\frac12)\) \(\approx\) \(0.975286 + 1.09794i\)
\(L(\frac{5}{2})\) 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 \( 1 + (371. + 403. i)T \)
good2 \( 1 + (-0.0760 + 0.131i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 2.82T + 125T^{2} \)
7 \( 1 + (1.28 + 2.22i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-16.0 - 27.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-2.60 + 4.50i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (4.88 - 8.46i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-4.30 + 7.46i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (65.0 - 112. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-129. - 224. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (7.00 + 12.1i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (191. - 332. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-225. - 390. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 356.T + 7.95e4T^{2} \)
47 \( 1 + (114. + 198. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 369.T + 1.48e5T^{2} \)
59 \( 1 + 24.2T + 2.05e5T^{2} \)
61 \( 1 + (-245. + 425. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (165. + 286. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (-341. + 592. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (35.6 + 61.7i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (211. - 365. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 624.T + 7.04e5T^{2} \)
97 \( 1 + (-103. + 179. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−12.15678198005593621626068272353, −11.48358614118759883213669916785, −10.42017791503690490001973967924, −9.395883341707011008351929350787, −8.064253546088577873009335318482, −7.08431201466880055052275778537, −6.17557801734679375190985349671, −4.70604777653714062401676722202, −3.41336651876617934366687133602, −1.74657424964941678527220721244, 0.67346011792143538327899842996, 2.27749044056768021981229097813, 4.24445859332241423638538783043, 5.74137166864957418453523581619, 6.18656893821162533300498416209, 7.42618678403027486575883503218, 8.913320235275961867447127785624, 9.964015355455310376550103549165, 10.78958614637737853425266695257, 11.60657165701488918596649325773

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