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Properties

Label 2-800-40.3-c1-0-7
Degree $2$
Conductor $800$
Sign $0.973 + 0.229i$
Analytic cond. $6.38803$
Root an. cond. $2.52745$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 2i)7-s − 3i·9-s − 2·11-s + (4 − 4i)13-s + 6i·19-s + (6 − 6i)23-s + (8 + 8i)37-s + 2·41-s + (2 + 2i)47-s + i·49-s + (4 − 4i)53-s − 14i·59-s + (6 − 6i)63-s + (−4 − 4i)77-s − 9·81-s + ⋯
L(s)  = 1  + (0.755 + 0.755i)7-s − i·9-s − 0.603·11-s + (1.10 − 1.10i)13-s + 1.37i·19-s + (1.25 − 1.25i)23-s + (1.31 + 1.31i)37-s + 0.312·41-s + (0.291 + 0.291i)47-s + 0.142i·49-s + (0.549 − 0.549i)53-s − 1.82i·59-s + (0.755 − 0.755i)63-s + (−0.455 − 0.455i)77-s − 81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(6.38803\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70486 - 0.198504i\)
\(L(\frac12)\) \(\approx\) \(1.70486 - 0.198504i\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT^{2} \)
7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-4 + 4i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 + (-6 + 6i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-8 - 8i)T + 37iT^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-2 - 2i)T + 47iT^{2} \)
53 \( 1 + (-4 + 4i)T - 53iT^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 14iT - 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−10.31692645994988297853290770243, −9.322747909375567810591850322028, −8.352330440303573290673536968148, −8.022280532909421398879180506484, −6.59036141897770206950397559466, −5.83105032034698313162827305544, −4.97805231081324452132162685448, −3.69687617049945404024994599216, −2.66930142592562766427950427273, −1.08982542230238577860434753361, 1.28824487055594822684623802319, 2.59570924697225173048039082859, 4.06899041323149409138534872621, 4.82556425173307612849732486746, 5.78775753037749034860274009525, 7.13069505742271052076652375419, 7.56461365013197317313338967011, 8.634438474652442509829578885489, 9.333467971837053596630541542261, 10.58599868625044597448294918682

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