↑

Properties

Label 2-20e2-5.4-c5-0-8
Degree $2$
Conductor $400$
Sign $-0.447 + 0.894i$
Analytic cond. $64.1535$
Root an. cond. $8.00958$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 24i·3-s + 172i·7-s − 333·9-s − 132·11-s + 946i·13-s − 222i·17-s + 500·19-s − 4.12e3·21-s + 3.56e3i·23-s − 2.16e3i·27-s − 2.19e3·29-s − 2.31e3·31-s − 3.16e3i·33-s − 1.12e4i·37-s − 2.27e4·39-s + ⋯
L(s)  = 1  + 1.53i·3-s + 1.32i·7-s − 1.37·9-s − 0.328·11-s + 1.55i·13-s − 0.186i·17-s + 0.317·19-s − 2.04·21-s + 1.40i·23-s − 0.570i·27-s − 0.483·29-s − 0.432·31-s − 0.506i·33-s − 1.35i·37-s − 2.39·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(64.1535\)
Root analytic conductor: \(8.00958\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :5/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.330398549\)
\(L(\frac12)\) \(\approx\) \(1.330398549\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - 24iT - 243T^{2} \)
7 \( 1 - 172iT - 1.68e4T^{2} \)
11 \( 1 + 132T + 1.61e5T^{2} \)
13 \( 1 - 946iT - 3.71e5T^{2} \)
17 \( 1 + 222iT - 1.41e6T^{2} \)
19 \( 1 - 500T + 2.47e6T^{2} \)
23 \( 1 - 3.56e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.19e3T + 2.05e7T^{2} \)
31 \( 1 + 2.31e3T + 2.86e7T^{2} \)
37 \( 1 + 1.12e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.24e3T + 1.15e8T^{2} \)
43 \( 1 - 2.06e4iT - 1.47e8T^{2} \)
47 \( 1 + 6.58e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.10e4iT - 4.18e8T^{2} \)
59 \( 1 - 7.98e3T + 7.14e8T^{2} \)
61 \( 1 - 1.66e4T + 8.44e8T^{2} \)
67 \( 1 + 1.80e3iT - 1.35e9T^{2} \)
71 \( 1 - 2.45e4T + 1.80e9T^{2} \)
73 \( 1 + 2.04e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.62e4T + 3.07e9T^{2} \)
83 \( 1 + 5.15e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.10e5T + 5.58e9T^{2} \)
97 \( 1 + 7.83e4iT - 8.58e9T^{2} \)
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

−11.20126901418439867195428815207, −9.935665805690466314738188326736, −9.260495622751062300999562447070, −8.838173691090341364202998843199, −7.43964408757535550862562883538, −5.97868768977696243609796420838, −5.22499422355608568090279560161, −4.27308816015194153223599460256, −3.22064386513993205444521383317, −1.97227214758327997329873411582, 0.36712056391158025581312131368, 1.03103811712080657343912961905, 2.38211939212763706650181030381, 3.61203156203784594417808070061, 5.12393732157512081686439900406, 6.28528227361147548101432668806, 7.14264599034062332950794760910, 7.79526988517850202405748539369, 8.504310067211096401092354883041, 10.11774491610350593784733003098

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