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Properties

Label 2-800-40.19-c4-0-39
Degree $2$
Conductor $800$
Sign $0.708 + 0.705i$
Analytic cond. $82.6959$
Root an. cond. $9.09373$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.715i·3-s − 25.0·7-s + 80.4·9-s + 156.·11-s − 301.·13-s − 123. i·17-s + 322.·19-s + 17.9i·21-s − 45.9·23-s − 115. i·27-s + 1.08e3i·29-s − 1.62e3i·31-s − 112. i·33-s + 895.·37-s + 215. i·39-s + ⋯
L(s)  = 1  − 0.0795i·3-s − 0.512·7-s + 0.993·9-s + 1.29·11-s − 1.78·13-s − 0.427i·17-s + 0.892·19-s + 0.0407i·21-s − 0.0867·23-s − 0.158i·27-s + 1.28i·29-s − 1.69i·31-s − 0.102i·33-s + 0.653·37-s + 0.141i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.708 + 0.705i$
Analytic conductor: \(82.6959\)
Root analytic conductor: \(9.09373\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :2),\ 0.708 + 0.705i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.034688947\)
\(L(\frac12)\) \(\approx\) \(2.034688947\)
\(L(3)\) 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 \( 1 + 0.715iT - 81T^{2} \)
7 \( 1 + 25.0T + 2.40e3T^{2} \)
11 \( 1 - 156.T + 1.46e4T^{2} \)
13 \( 1 + 301.T + 2.85e4T^{2} \)
17 \( 1 + 123. iT - 8.35e4T^{2} \)
19 \( 1 - 322.T + 1.30e5T^{2} \)
23 \( 1 + 45.9T + 2.79e5T^{2} \)
29 \( 1 - 1.08e3iT - 7.07e5T^{2} \)
31 \( 1 + 1.62e3iT - 9.23e5T^{2} \)
37 \( 1 - 895.T + 1.87e6T^{2} \)
41 \( 1 + 1.09e3T + 2.82e6T^{2} \)
43 \( 1 - 950. iT - 3.41e6T^{2} \)
47 \( 1 - 1.34e3T + 4.87e6T^{2} \)
53 \( 1 + 709.T + 7.89e6T^{2} \)
59 \( 1 - 4.46e3T + 1.21e7T^{2} \)
61 \( 1 - 933. iT - 1.38e7T^{2} \)
67 \( 1 - 4.01e3iT - 2.01e7T^{2} \)
71 \( 1 - 3.55e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.29e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.14e4iT - 3.89e7T^{2} \)
83 \( 1 + 6.58e3iT - 4.74e7T^{2} \)
89 \( 1 - 6.28e3T + 6.27e7T^{2} \)
97 \( 1 + 1.49e4iT - 8.85e7T^{2} \)
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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

−9.707899746686926118435890858557, −8.978473012318413468621799929281, −7.54981710185149405398129821460, −7.13992860498343973061974818362, −6.21713238581939435189442469016, −5.00772284176586408764795866956, −4.17019143576564062745228996059, −3.05424301968779521143965222112, −1.81282084019503347503785846276, −0.58380119148057025799174220237, 0.906379640922925604765481294656, 2.11864254438893077523348528485, 3.41980841816687339471706027548, 4.34583874277681516729355595229, 5.26167498202264146752330379555, 6.54908155763620292019067944906, 7.07026288552170538815832849392, 7.996675253700443955113106551766, 9.250735943107217486552412427903, 9.733612844737845676560858383991

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