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Properties

Label 2-800-5.3-c2-0-7
Degree $2$
Conductor $800$
Sign $-0.973 - 0.229i$
Analytic cond. $21.7984$
Root an. cond. $4.66887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.89 + 2.89i)3-s + (−5.86 + 5.86i)7-s + 7.76i·9-s − 7.84·11-s + (15.5 + 15.5i)13-s + (−13.3 + 13.3i)17-s − 21.1i·19-s − 33.9·21-s + (−2.28 − 2.28i)23-s + (3.58 − 3.58i)27-s + 7.68i·29-s − 32.1·31-s + (−22.7 − 22.7i)33-s + (−38.2 + 38.2i)37-s + 90.0i·39-s + ⋯
L(s)  = 1  + (0.965 + 0.965i)3-s + (−0.838 + 0.838i)7-s + 0.862i·9-s − 0.713·11-s + (1.19 + 1.19i)13-s + (−0.788 + 0.788i)17-s − 1.11i·19-s − 1.61·21-s + (−0.0994 − 0.0994i)23-s + (0.132 − 0.132i)27-s + 0.265i·29-s − 1.03·31-s + (−0.688 − 0.688i)33-s + (−1.03 + 1.03i)37-s + 2.30i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(21.7984\)
Root analytic conductor: \(4.66887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :1),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.543283874\)
\(L(\frac12)\) \(\approx\) \(1.543283874\)
\(L(2)\) 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 \( 1 + (-2.89 - 2.89i)T + 9iT^{2} \)
7 \( 1 + (5.86 - 5.86i)T - 49iT^{2} \)
11 \( 1 + 7.84T + 121T^{2} \)
13 \( 1 + (-15.5 - 15.5i)T + 169iT^{2} \)
17 \( 1 + (13.3 - 13.3i)T - 289iT^{2} \)
19 \( 1 + 21.1iT - 361T^{2} \)
23 \( 1 + (2.28 + 2.28i)T + 529iT^{2} \)
29 \( 1 - 7.68iT - 841T^{2} \)
31 \( 1 + 32.1T + 961T^{2} \)
37 \( 1 + (38.2 - 38.2i)T - 1.36e3iT^{2} \)
41 \( 1 - 20.8T + 1.68e3T^{2} \)
43 \( 1 + (57.9 + 57.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (35.1 - 35.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (-22.8 - 22.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 69.7iT - 3.48e3T^{2} \)
61 \( 1 + 11.9T + 3.72e3T^{2} \)
67 \( 1 + (-27.5 + 27.5i)T - 4.48e3iT^{2} \)
71 \( 1 - 52.2T + 5.04e3T^{2} \)
73 \( 1 + (-4.73 - 4.73i)T + 5.32e3iT^{2} \)
79 \( 1 - 31.3iT - 6.24e3T^{2} \)
83 \( 1 + (33.3 + 33.3i)T + 6.88e3iT^{2} \)
89 \( 1 + 0.623iT - 7.92e3T^{2} \)
97 \( 1 + (-83.1 + 83.1i)T - 9.40e3iT^{2} \)
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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

−10.34346200550641384776583865819, −9.359060656305999536211304717657, −8.881190493281331302589660786437, −8.386089649306707975642055786323, −6.92323988426227090245822706328, −6.12039660629517101727561629421, −4.91700164990254107820516495365, −3.90807886509380556466593241067, −3.10554371465301902072980676479, −2.05820688819565461117188818901, 0.43100822835029356824889450157, 1.82405981814815564689383571888, 3.08902258383451263072019061538, 3.72239273408991508539158729637, 5.33089625237616585082865931384, 6.42235516839251423453679768353, 7.21267928499621138125161449665, 7.970538449355996947388358983675, 8.561588128522392434483656987111, 9.643747076161551546068175051845

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