↑

Properties

Label 2-200-40.27-c3-0-43
Degree $2$
Conductor $200$
Sign $0.377 + 0.926i$
Analytic cond. $11.8003$
Root an. cond. $3.43516$
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  + (−0.589 + 2.76i)2-s + (6.56 − 6.56i)3-s + (−7.30 − 3.26i)4-s + (14.2 + 22.0i)6-s + (8.83 − 8.83i)7-s + (13.3 − 18.2i)8-s − 59.1i·9-s − 28.5·11-s + (−69.3 + 26.5i)12-s + (−22.4 − 22.4i)13-s + (19.2 + 29.6i)14-s + (42.6 + 47.6i)16-s + (15.7 + 15.7i)17-s + (163. + 34.8i)18-s + 4.45i·19-s + ⋯
L(s)  = 1  + (−0.208 + 0.978i)2-s + (1.26 − 1.26i)3-s + (−0.912 − 0.407i)4-s + (0.971 + 1.49i)6-s + (0.476 − 0.476i)7-s + (0.589 − 0.807i)8-s − 2.18i·9-s − 0.781·11-s + (−1.66 + 0.637i)12-s + (−0.478 − 0.478i)13-s + (0.366 + 0.565i)14-s + (0.667 + 0.744i)16-s + (0.224 + 0.224i)17-s + (2.14 + 0.456i)18-s + 0.0538i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.926i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.377 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $0.377 + 0.926i$
Analytic conductor: \(11.8003\)
Root analytic conductor: \(3.43516\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 200,\ (\ :3/2),\ 0.377 + 0.926i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.61310 - 1.08468i\)
\(L(\frac12)\) \(\approx\) \(1.61310 - 1.08468i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.589 - 2.76i)T \)
5 \( 1 \)
good3 \( 1 + (-6.56 + 6.56i)T - 27iT^{2} \)
7 \( 1 + (-8.83 + 8.83i)T - 343iT^{2} \)
11 \( 1 + 28.5T + 1.33e3T^{2} \)
13 \( 1 + (22.4 + 22.4i)T + 2.19e3iT^{2} \)
17 \( 1 + (-15.7 - 15.7i)T + 4.91e3iT^{2} \)
19 \( 1 - 4.45iT - 6.85e3T^{2} \)
23 \( 1 + (116. + 116. i)T + 1.21e4iT^{2} \)
29 \( 1 - 132.T + 2.43e4T^{2} \)
31 \( 1 + 48.9iT - 2.97e4T^{2} \)
37 \( 1 + (-280. + 280. i)T - 5.06e4iT^{2} \)
41 \( 1 - 148.T + 6.89e4T^{2} \)
43 \( 1 + (-192. + 192. i)T - 7.95e4iT^{2} \)
47 \( 1 + (152. - 152. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-225. - 225. i)T + 1.48e5iT^{2} \)
59 \( 1 + 102. iT - 2.05e5T^{2} \)
61 \( 1 - 310. iT - 2.26e5T^{2} \)
67 \( 1 + (-560. - 560. i)T + 3.00e5iT^{2} \)
71 \( 1 - 4.97iT - 3.57e5T^{2} \)
73 \( 1 + (-106. + 106. i)T - 3.89e5iT^{2} \)
79 \( 1 - 742.T + 4.93e5T^{2} \)
83 \( 1 + (692. - 692. i)T - 5.71e5iT^{2} \)
89 \( 1 - 439. iT - 7.04e5T^{2} \)
97 \( 1 + (872. + 872. i)T + 9.12e5iT^{2} \)
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

−12.39495407802482982131854709366, −10.55693252857554568053490970987, −9.453366469612346788869406206959, −8.262669094912110906633711822028, −7.86851720435728845126084221415, −7.03926834820583792718506015697, −5.85010812571858338527702528162, −4.19583220418931447510350447158, −2.47117848536381848616297329274, −0.793996657200508848833594336127, 2.13217061128139801130166290092, 3.11282683035970001842094148824, 4.33256848097478300886477557088, 5.20933573968693888969708463767, 7.83058360803745581111004062444, 8.420640763822908201585910505917, 9.521867059771926009636619495710, 9.963803766808679168290416418346, 11.02244932378968368023218629783, 11.97989969755149622228901164477

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