Properties

Label 2-40e2-20.19-c2-0-41
Degree $2$
Conductor $1600$
Sign $-0.447 + 0.894i$
Analytic cond. $43.5968$
Root an. cond. $6.60279$
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  − 2.35·3-s − 5.25·7-s − 3.47·9-s + 19.9i·11-s − 8.47i·13-s + 11.8i·17-s + 15.2i·19-s + 12.3·21-s + 0.555·23-s + 29.3·27-s − 10.9·29-s + 8.29i·31-s − 46.8i·33-s + 18.3i·37-s + 19.9i·39-s + ⋯
L(s)  = 1  − 0.783·3-s − 0.751·7-s − 0.385·9-s + 1.81i·11-s − 0.651i·13-s + 0.699i·17-s + 0.800i·19-s + 0.588·21-s + 0.0241·23-s + 1.08·27-s − 0.377·29-s + 0.267i·31-s − 1.41i·33-s + 0.496i·37-s + 0.510i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(43.5968\)
Root analytic conductor: \(6.60279\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1208927470\)
\(L(\frac12)\) \(\approx\) \(0.1208927470\)
\(L(2)\) 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 \( 1 + 2.35T + 9T^{2} \)
7 \( 1 + 5.25T + 49T^{2} \)
11 \( 1 - 19.9iT - 121T^{2} \)
13 \( 1 + 8.47iT - 169T^{2} \)
17 \( 1 - 11.8iT - 289T^{2} \)
19 \( 1 - 15.2iT - 361T^{2} \)
23 \( 1 - 0.555T + 529T^{2} \)
29 \( 1 + 10.9T + 841T^{2} \)
31 \( 1 - 8.29iT - 961T^{2} \)
37 \( 1 - 18.3iT - 1.36e3T^{2} \)
41 \( 1 + 14.5T + 1.68e3T^{2} \)
43 \( 1 - 22.2T + 1.84e3T^{2} \)
47 \( 1 - 53.3T + 2.20e3T^{2} \)
53 \( 1 + 66.3iT - 2.80e3T^{2} \)
59 \( 1 + 17.4iT - 3.48e3T^{2} \)
61 \( 1 + 90.1T + 3.72e3T^{2} \)
67 \( 1 + 50.2T + 4.48e3T^{2} \)
71 \( 1 - 80.7iT - 5.04e3T^{2} \)
73 \( 1 - 5.55iT - 5.32e3T^{2} \)
79 \( 1 + 13.8iT - 6.24e3T^{2} \)
83 \( 1 + 76.2T + 6.88e3T^{2} \)
89 \( 1 - 111.T + 7.92e3T^{2} \)
97 \( 1 + 92.8iT - 9.40e3T^{2} \)
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

−9.070512716132164315560795658512, −8.092246416908693251083448066931, −7.24743785646176384664297058809, −6.45411990650407540607555000211, −5.74199142296887533561408963326, −4.92454517266979640400249112472, −3.95612832871877933676695703107, −2.84997886081603924204572882148, −1.61494612276011370827615963154, −0.04822195018889891199494455091, 0.846857178143895243824463394306, 2.64796134575275302372323395604, 3.42025020627999688450710438469, 4.59476882922638818861633487194, 5.63216519101331197425693810533, 6.10683089144868530046375040378, 6.86529402065793921773056477753, 7.85397852219475848873820971395, 9.040826242110164030151816010093, 9.145730297342172928112897504599

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