Properties

Label 2-201-67.29-c3-0-26
Degree $2$
Conductor $201$
Sign $0.505 + 0.862i$
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  + (−1.18 + 2.05i)2-s + 3·3-s + (1.19 + 2.07i)4-s − 11.7·5-s + (−3.55 + 6.15i)6-s + (−6.15 − 10.6i)7-s − 24.6·8-s + 9·9-s + (13.9 − 24.1i)10-s + (−9.68 − 16.7i)11-s + (3.59 + 6.22i)12-s + (31.4 − 54.5i)13-s + 29.1·14-s − 35.2·15-s + (19.5 − 33.8i)16-s + (14.4 − 24.9i)17-s + ⋯
L(s)  = 1  + (−0.418 + 0.724i)2-s + 0.577·3-s + (0.149 + 0.259i)4-s − 1.05·5-s + (−0.241 + 0.418i)6-s + (−0.332 − 0.575i)7-s − 1.08·8-s + 0.333·9-s + (0.440 − 0.762i)10-s + (−0.265 − 0.459i)11-s + (0.0864 + 0.149i)12-s + (0.671 − 1.16i)13-s + 0.556·14-s − 0.607·15-s + (0.305 − 0.529i)16-s + (0.205 − 0.356i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.505 + 0.862i$
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: \((2,\ 201,\ (\ :3/2),\ 0.505 + 0.862i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.671673 - 0.384814i\)
\(L(\frac12)\) \(\approx\) \(0.671673 - 0.384814i\)
\(L(\frac{5}{2})\) 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 \( 1 + (502. + 220. i)T \)
good2 \( 1 + (1.18 - 2.05i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 11.7T + 125T^{2} \)
7 \( 1 + (6.15 + 10.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (9.68 + 16.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-31.4 + 54.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-14.4 + 24.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-15.2 + 26.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (29.9 - 51.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (51.4 + 89.0i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (33.2 + 57.6i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-11.4 + 19.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (57.9 + 100. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 - 160.T + 7.95e4T^{2} \)
47 \( 1 + (45.1 + 78.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 151.T + 1.48e5T^{2} \)
59 \( 1 + 235.T + 2.05e5T^{2} \)
61 \( 1 + (-241. + 418. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
71 \( 1 + (257. + 446. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + (439. - 760. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (185. + 321. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (414. - 718. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.47e3T + 7.04e5T^{2} \)
97 \( 1 + (515. - 892. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−11.82610137164522400151939114034, −10.86719860398619839323398725974, −9.598570285741992312970254982336, −8.424008183903633475766539020489, −7.82677378998326002811478949838, −7.10839060443608980808933025276, −5.75811275142748024554563250789, −3.87437312662759020825200068080, −3.04669587282651918563137982458, −0.35333522213816506594212567904, 1.63203056661629341230413424407, 3.01164232841589464114178949042, 4.21438924816046408980838598004, 5.97795783961772991986836991552, 7.21070744878165716205036842764, 8.455621958454212218564705529975, 9.202324536869211562773775507052, 10.19754015417019076535795450686, 11.23301972541653963575746568680, 11.99981218008937653922292324710

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