Properties

Label 2-800-5.3-c0-0-1
Degree $2$
Conductor $800$
Sign $0.973 + 0.229i$
Analytic cond. $0.399252$
Root an. cond. $0.631863$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·9-s + (1 + i)13-s + (1 − i)17-s + (−1 + i)37-s + i·49-s + (−1 − i)53-s + (−1 − i)73-s − 81-s + (−1 + i)97-s − 2·101-s + 2i·109-s + (1 + i)113-s + (1 − i)117-s + ⋯
L(s)  = 1  i·9-s + (1 + i)13-s + (1 − i)17-s + (−1 + i)37-s + i·49-s + (−1 − i)53-s + (−1 − i)73-s − 81-s + (−1 + i)97-s − 2·101-s + 2i·109-s + (1 + i)113-s + (1 − i)117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :0),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.017967173\)
\(L(\frac12)\) \(\approx\) \(1.017967173\)
\(L(1)\) 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 \( 1 + iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−10.37508677141032781256638495775, −9.473674477475567967323353023605, −8.917100478281945147692390632369, −7.902183045738170981526358898506, −6.85352309348448355130523764019, −6.20513009170500104962279076153, −5.09689581356146083295485265421, −3.94475876647355079220750533822, −3.05444402249148542367272708883, −1.36895002068491677992741418241, 1.58849934508451499306883648860, 3.04004687598663522254644475155, 4.06483515617659980911558398079, 5.35050401038494667986764952731, 5.92165793282480784092446772764, 7.17792146549312341326590544182, 8.077543216265555129976011944201, 8.572212431423190890905315262018, 9.822517529218652472810256623789, 10.59992151531658934819817186097

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