↑

Properties

Label 2-201-67.30-c2-0-18
Degree $2$
Conductor $201$
Sign $0.0734 + 0.997i$
Analytic cond. $5.47685$
Root an. cond. $2.34026$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.98 − 1.72i)2-s + 1.73i·3-s + (3.95 − 6.85i)4-s − 6.27i·5-s + (2.98 + 5.17i)6-s + (−1.92 − 1.10i)7-s − 13.5i·8-s − 2.99·9-s + (−10.8 − 18.7i)10-s + (0.342 + 0.197i)11-s + (11.8 + 6.85i)12-s + (7.58 − 4.37i)13-s − 7.66·14-s + 10.8·15-s + (−7.52 − 13.0i)16-s + (13.4 + 23.2i)17-s + ⋯
L(s)  = 1  + (1.49 − 0.863i)2-s + 0.577i·3-s + (0.989 − 1.71i)4-s − 1.25i·5-s + (0.498 + 0.863i)6-s + (−0.274 − 0.158i)7-s − 1.69i·8-s − 0.333·9-s + (−1.08 − 1.87i)10-s + (0.0311 + 0.0179i)11-s + (0.989 + 0.571i)12-s + (0.583 − 0.336i)13-s − 0.547·14-s + 0.724·15-s + (−0.470 − 0.814i)16-s + (0.789 + 1.36i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0734 + 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0734 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(201\)    =    \(3 \cdot 67\)
Sign: $0.0734 + 0.997i$
Analytic conductor: \(5.47685\)
Root analytic conductor: \(2.34026\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{201} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 201,\ (\ :1),\ 0.0734 + 0.997i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.40369 - 2.23310i\)
\(L(\frac12)\) \(\approx\) \(2.40369 - 2.23310i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
67 \( 1 + (48.2 + 46.4i)T \)
good2 \( 1 + (-2.98 + 1.72i)T + (2 - 3.46i)T^{2} \)
5 \( 1 + 6.27iT - 25T^{2} \)
7 \( 1 + (1.92 + 1.10i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.342 - 0.197i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-7.58 + 4.37i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-13.4 - 23.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-5.15 - 8.92i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (1.08 + 1.88i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (11.9 - 20.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (46.3 + 26.7i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-20.1 - 34.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-20.0 - 11.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + 26.9iT - 1.84e3T^{2} \)
47 \( 1 + (10.2 - 17.7i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 77.9iT - 2.80e3T^{2} \)
59 \( 1 + 69.2T + 3.48e3T^{2} \)
61 \( 1 + (-24.5 + 14.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
71 \( 1 + (-22.0 + 38.2i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (-26.8 - 46.4i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-108. - 62.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (49.1 + 85.1i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 5.71T + 7.92e3T^{2} \)
97 \( 1 + (23.0 - 13.2i)T + (4.70e3 - 8.14e3i)T^{2} \)
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

−12.35064850912204119639541262690, −11.18569304027908120770503234623, −10.37034268986062554681544490105, −9.305460850721747815914882006572, −8.079626116527604091236786131575, −6.04834457035996532722593009127, −5.32012434002905746405064209476, −4.21740644208003269209550115963, −3.39227963642098977399506330720, −1.46756598811568121996575283252, 2.69834990813542834825057269833, 3.64684154132541083225452029948, 5.26685636563976103330872038568, 6.26290432595597536943042461624, 7.04844689870014534719119866892, 7.70703835362463637559734191737, 9.393930239244742386256214465717, 10.99754549068089247329982826586, 11.75032675181502542581519119510, 12.74360231613542495469050096933

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