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Properties

Label 2-800-40.19-c0-0-0
Degree $2$
Conductor $800$
Sign $0.447 - 0.894i$
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·3-s + 11-s + i·17-s − 19-s + i·27-s + i·33-s − 41-s − 2i·43-s − 49-s − 51-s − i·57-s + 2·59-s − i·67-s − i·73-s − 81-s + ⋯
L(s)  = 1  + i·3-s + 11-s + i·17-s − 19-s + i·27-s + i·33-s − 41-s − 2i·43-s − 49-s − 51-s − i·57-s + 2·59-s − i·67-s − i·73-s − 81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024087124\)
\(L(\frac12)\) \(\approx\) \(1.024087124\)
\(L(1)\) 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 \( 1 - iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + 2iT - T^{2} \)
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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

−10.48192462057179431669655267948, −9.888970010120026042719453466201, −8.974148495807358145771356534967, −8.360859442106513473343411294213, −7.05938144605083630950688159385, −6.24893824838174372412763924405, −5.13329225274981955757854630763, −4.14552487292280802101351485904, −3.54888068522024869069231136553, −1.84068117969644956727528700491, 1.27683073765345448383954374058, 2.49032747626348307530879777677, 3.89058780516858376452091421139, 4.96042355551600179416120835091, 6.30808991530566763121933921838, 6.76979535282880298690896417891, 7.66976433374811804530926445176, 8.524658810398930347786710734400, 9.433726876346548985088267965846, 10.26471179692433523487791927714

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