Properties

Label 2-800-4.3-c2-0-27
Degree $2$
Conductor $800$
Sign $0.707 + 0.707i$
Analytic cond. $21.7984$
Root an. cond. $4.66887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.30i·3-s + 0.206i·7-s − 19.1·9-s − 15.0i·11-s − 11.6·13-s + 18.1·17-s − 19.3i·19-s − 1.09·21-s − 27.2i·23-s − 53.6i·27-s − 44.4·29-s − 20.3i·31-s + 79.6·33-s + 18.1·37-s − 62.0i·39-s + ⋯
L(s)  = 1  + 1.76i·3-s + 0.0295i·7-s − 2.12·9-s − 1.36i·11-s − 0.899·13-s + 1.07·17-s − 1.02i·19-s − 0.0521·21-s − 1.18i·23-s − 1.98i·27-s − 1.53·29-s − 0.657i·31-s + 2.41·33-s + 0.489·37-s − 1.59i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(21.7984\)
Root analytic conductor: \(4.66887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8495160090\)
\(L(\frac12)\) \(\approx\) \(0.8495160090\)
\(L(2)\) 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 \( 1 - 5.30iT - 9T^{2} \)
7 \( 1 - 0.206iT - 49T^{2} \)
11 \( 1 + 15.0iT - 121T^{2} \)
13 \( 1 + 11.6T + 169T^{2} \)
17 \( 1 - 18.1T + 289T^{2} \)
19 \( 1 + 19.3iT - 361T^{2} \)
23 \( 1 + 27.2iT - 529T^{2} \)
29 \( 1 + 44.4T + 841T^{2} \)
31 \( 1 + 20.3iT - 961T^{2} \)
37 \( 1 - 18.1T + 1.36e3T^{2} \)
41 \( 1 + 32.3T + 1.68e3T^{2} \)
43 \( 1 + 4.06iT - 1.84e3T^{2} \)
47 \( 1 + 5.37iT - 2.20e3T^{2} \)
53 \( 1 + 79.1T + 2.80e3T^{2} \)
59 \( 1 - 83.3iT - 3.48e3T^{2} \)
61 \( 1 + 36.7T + 3.72e3T^{2} \)
67 \( 1 - 4.51iT - 4.48e3T^{2} \)
71 \( 1 + 41.6iT - 5.04e3T^{2} \)
73 \( 1 - 41.5T + 5.32e3T^{2} \)
79 \( 1 + 15.5iT - 6.24e3T^{2} \)
83 \( 1 + 50.9iT - 6.88e3T^{2} \)
89 \( 1 + 10.8T + 7.92e3T^{2} \)
97 \( 1 - 12.1T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.949941893220896879927322790327, −9.237622854895694434901319752198, −8.584969810895850241842735302380, −7.55594648766376562086133899181, −6.10797151853311187905249350782, −5.34559874009358095191875622366, −4.52467214719913719560104950485, −3.53435357688124898477127901244, −2.70905900288798372574988483851, −0.28598036489030180847748494044, 1.39946803803984614867414536093, 2.14888236591819552324899314748, 3.46868957315388694714585776217, 5.07120599242926733142273387454, 5.92231434120680956082472172114, 6.96889489303890563426014459686, 7.55600871711364804729649121283, 8.009247094928790964246259837808, 9.346389061779468604801147616785, 10.03797439772891393557691478610

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