↑

Properties

Label 2-40e2-8.3-c2-0-52
Degree $2$
Conductor $1600$
Sign $0.707 + 0.707i$
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  + 1.08·3-s + 4.41i·7-s − 7.83·9-s + 17.7·11-s − 15.5i·13-s − 10.8·17-s + 11.2·19-s + 4.77i·21-s − 25.2i·23-s − 18.1·27-s + 31.1i·29-s − 11.1i·31-s + 19.1·33-s − 57.5i·37-s − 16.8i·39-s + ⋯
L(s)  = 1  + 0.360·3-s + 0.630i·7-s − 0.870·9-s + 1.61·11-s − 1.19i·13-s − 0.637·17-s + 0.592·19-s + 0.227i·21-s − 1.09i·23-s − 0.673·27-s + 1.07i·29-s − 0.360i·31-s + 0.580·33-s − 1.55i·37-s − 0.431i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(43.5968\)
Root analytic conductor: \(6.60279\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :1),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.105362810\)
\(L(\frac12)\) \(\approx\) \(2.105362810\)
\(L(2)\) 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 \( 1 - 1.08T + 9T^{2} \)
7 \( 1 - 4.41iT - 49T^{2} \)
11 \( 1 - 17.7T + 121T^{2} \)
13 \( 1 + 15.5iT - 169T^{2} \)
17 \( 1 + 10.8T + 289T^{2} \)
19 \( 1 - 11.2T + 361T^{2} \)
23 \( 1 + 25.2iT - 529T^{2} \)
29 \( 1 - 31.1iT - 841T^{2} \)
31 \( 1 + 11.1iT - 961T^{2} \)
37 \( 1 + 57.5iT - 1.36e3T^{2} \)
41 \( 1 - 40.8T + 1.68e3T^{2} \)
43 \( 1 + 56.9T + 1.84e3T^{2} \)
47 \( 1 - 5.91iT - 2.20e3T^{2} \)
53 \( 1 - 24.2iT - 2.80e3T^{2} \)
59 \( 1 - 2.61T + 3.48e3T^{2} \)
61 \( 1 - 0.449iT - 3.72e3T^{2} \)
67 \( 1 - 72.0T + 4.48e3T^{2} \)
71 \( 1 + 120. iT - 5.04e3T^{2} \)
73 \( 1 - 124.T + 5.32e3T^{2} \)
79 \( 1 - 29.6iT - 6.24e3T^{2} \)
83 \( 1 - 141.T + 6.88e3T^{2} \)
89 \( 1 - 39.6T + 7.92e3T^{2} \)
97 \( 1 + 98.1T + 9.40e3T^{2} \)
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

−8.989840757015337651982712720071, −8.546122454070899341738182858960, −7.63369846705852789511441748497, −6.61607622888304379645178659247, −5.89903242067339964911901854982, −5.07730224762403072051445555129, −3.88726795478234420624223089543, −3.05075535480606707328820489834, −2.08420701411001732157925075419, −0.62231744728085753394412077958, 1.08075736742077607844925469093, 2.18403154174327745552277536838, 3.50648376731786877938011828850, 4.07614645003659441993421399713, 5.13208652997818978312600210464, 6.34492471428555322114872706868, 6.77923342665734443895444500823, 7.77839005797044330457327029686, 8.632752304546797289013495588051, 9.334914409093342288726430452945

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