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  =  \((2,\ 800,\ (\ :5/2),\ 0.894 - 0.447i)\)
\(L(3)\)  \(\approx\)  \(2.366805367\)
\(L(\frac12)\)  \(\approx\)  \(2.366805367\)
\(L(\frac{7}{2})\)   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

−9.490538319044207041817344594487, −9.097411148008982936456625408406, −7.72885481782015995278886808443, −7.09856804371064434662156661704, −6.06417016047626141266947412115, −4.88539348994232004145439305586, −4.06574541267715744060213379862, −3.52270301494093885297101614858, −1.88403841735864449719640416385, −0.71153054387486928716542038399, 0.71962466511199404234081476054, 1.82674862289652140693352811158, 2.64575071258979439869286728521, 3.99412607673623166964252787070, 5.09478022437639887587343039248, 6.24403911066181939868423854822, 6.65553626169735182784301470583, 7.82663758824097923669667712184, 8.481012887961840846287171249618, 9.353973373998657656542316018854

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