Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{2} $
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

\( d \)  =  \(2\)
\( N \)  =  \(800\)    =    \(2^{5} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.894 - 0.447i$
motivic weight  =  \(5\)
character  :  $\chi_{800} (449, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 800,\ (\ :5/2),\ 0.894 - 0.447i)\)
\(L(3)\)  \(\approx\)  \(1.862989546\)
\(L(\frac12)\)  \(\approx\)  \(1.862989546\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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