Properties

Degree $2$
Conductor $800$
Sign $0.894 + 0.447i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.4i·3-s − 138. i·7-s + 62.9·9-s − 259.·11-s − 154i·13-s + 178i·17-s + 965.·19-s + 1.86e3·21-s + 2.63e3i·23-s + 4.10e3i·27-s − 4.11e3·29-s − 3.15e3·31-s − 3.48e3i·33-s + 7.44e3i·37-s + 2.06e3·39-s + ⋯
L(s)  = 1  + 0.860i·3-s − 1.06i·7-s + 0.259·9-s − 0.646·11-s − 0.252i·13-s + 0.149i·17-s + 0.613·19-s + 0.920·21-s + 1.03i·23-s + 1.08i·27-s − 0.907·29-s − 0.590·31-s − 0.556i·33-s + 0.893i·37-s + 0.217·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Motivic weight: \(5\)
Character: $\chi_{800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.862989546\)
\(L(\frac12)\) \(\approx\) \(1.862989546\)
\(L(\frac{7}{2})\) 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 \( 1 - 13.4iT - 243T^{2} \)
7 \( 1 + 138. iT - 1.68e4T^{2} \)
11 \( 1 + 259.T + 1.61e5T^{2} \)
13 \( 1 + 154iT - 3.71e5T^{2} \)
17 \( 1 - 178iT - 1.41e6T^{2} \)
19 \( 1 - 965.T + 2.47e6T^{2} \)
23 \( 1 - 2.63e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.11e3T + 2.05e7T^{2} \)
31 \( 1 + 3.15e3T + 2.86e7T^{2} \)
37 \( 1 - 7.44e3iT - 6.93e7T^{2} \)
41 \( 1 - 7.27e3T + 1.15e8T^{2} \)
43 \( 1 + 1.79e4iT - 1.47e8T^{2} \)
47 \( 1 - 7.41e3iT - 2.29e8T^{2} \)
53 \( 1 + 3.22e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.40e4T + 7.14e8T^{2} \)
61 \( 1 - 2.67e4T + 8.44e8T^{2} \)
67 \( 1 + 4.98e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.41e4T + 1.80e9T^{2} \)
73 \( 1 - 1.85e4iT - 2.07e9T^{2} \)
79 \( 1 + 8.67e4T + 3.07e9T^{2} \)
83 \( 1 + 7.86e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.07e5T + 5.58e9T^{2} \)
97 \( 1 + 1.08e5iT - 8.58e9T^{2} \)
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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

−9.741728822724002482639123377304, −8.717644981670198552739618570204, −7.56546113518843757515473467671, −7.12834780681454575742228020301, −5.70523311621998698547455519608, −4.91786188293768492943996423392, −3.94266725546524276282007294041, −3.29034309820472129584872968806, −1.72378999548811139924873476545, −0.46733847554135373672318054086, 0.847397695707074942104226766243, 2.03988482455286953805762988253, 2.74817825128681814274466501333, 4.18525394758068224106471851620, 5.35273911334548315379133226390, 6.07839447024359736275730452210, 7.09996295984922741282603514617, 7.75136667190852024725624894208, 8.686208331975624092717733849034, 9.448632924088516516640784648179

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