Properties

Label 2-800-5.3-c0-0-0
Degree $2$
Conductor $800$
Sign $0.229 - 0.973i$
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  + (1 + i)3-s + (−1 + i)7-s + i·9-s − 2·21-s + (1 + i)23-s − 2i·29-s + (−1 − i)43-s + (1 − i)47-s i·49-s + (−1 − i)63-s + (−1 + i)67-s + 2i·69-s + 81-s + (−1 − i)83-s + (2 − 2i)87-s + ⋯
L(s)  = 1  + (1 + i)3-s + (−1 + i)7-s + i·9-s − 2·21-s + (1 + i)23-s − 2i·29-s + (−1 − i)43-s + (1 − i)47-s i·49-s + (−1 − i)63-s + (−1 + i)67-s + 2i·69-s + 81-s + (−1 − i)83-s + (2 − 2i)87-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.229 - 0.973i$
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: \((2,\ 800,\ (\ :0),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.203153927\)
\(L(\frac12)\) \(\approx\) \(1.203153927\)
\(L(1)\) 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 \( 1 + (-1 - i)T + iT^{2} \)
7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 + (-1 + i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.25418350284940333018242297933, −9.712318834154996844364446445339, −9.035576405082033067896923317804, −8.472426697580318999022771052851, −7.33705878273929921372252718273, −6.18178692110613313626959723916, −5.27022331743516471751462930084, −4.06560687717947572477017256657, −3.21439990882608355955517968670, −2.37520475548774116647543465138, 1.27596552012151008585594330858, 2.77337933298861100698574492004, 3.48879543380692432366847665689, 4.78912913688693580623298635546, 6.32662121816747505270602289693, 6.99530368165946003159275129257, 7.57101883992844295644297705453, 8.602158146563770624393973871475, 9.235042093885249695731507453882, 10.27436645597235270369076056873

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