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Properties

Label 1-800-800.381-r0-0-0
Degree $1$
Conductor $800$
Sign $0.331 - 0.943i$
Analytic cond. $3.71518$
Root an. cond. $3.71518$
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  + (0.891 − 0.453i)3-s − i·7-s + (0.587 − 0.809i)9-s + (−0.156 + 0.987i)11-s + (0.156 + 0.987i)13-s + (−0.309 − 0.951i)17-s + (0.453 − 0.891i)19-s + (−0.453 − 0.891i)21-s + (−0.587 − 0.809i)23-s + (0.156 − 0.987i)27-s + (0.891 − 0.453i)29-s + (0.309 + 0.951i)31-s + (0.309 + 0.951i)33-s + (0.987 − 0.156i)37-s + (0.587 + 0.809i)39-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)3-s − i·7-s + (0.587 − 0.809i)9-s + (−0.156 + 0.987i)11-s + (0.156 + 0.987i)13-s + (−0.309 − 0.951i)17-s + (0.453 − 0.891i)19-s + (−0.453 − 0.891i)21-s + (−0.587 − 0.809i)23-s + (0.156 − 0.987i)27-s + (0.891 − 0.453i)29-s + (0.309 + 0.951i)31-s + (0.309 + 0.951i)33-s + (0.987 − 0.156i)37-s + (0.587 + 0.809i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.331 - 0.943i$
Analytic conductor: \(3.71518\)
Root analytic conductor: \(3.71518\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (0:\ ),\ 0.331 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.616088455 - 1.145313394i\)
\(L(\frac12)\) \(\approx\) \(1.616088455 - 1.145313394i\)
\(L(1)\) \(\approx\) \(1.368080227 - 0.4345488752i\)
\(L(1)\) \(\approx\) \(1.368080227 - 0.4345488752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.891 - 0.453i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.156 + 0.987i)T \)
13 \( 1 + (0.156 + 0.987i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.453 - 0.891i)T \)
23 \( 1 + (-0.587 - 0.809i)T \)
29 \( 1 + (0.891 - 0.453i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.987 - 0.156i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.453 - 0.891i)T \)
59 \( 1 + (0.987 - 0.156i)T \)
61 \( 1 + (-0.987 - 0.156i)T \)
67 \( 1 + (-0.453 + 0.891i)T \)
71 \( 1 + (-0.951 - 0.309i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (-0.309 + 0.951i)T \)
83 \( 1 + (0.453 - 0.891i)T \)
89 \( 1 + (0.587 + 0.809i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.0598873932975061520962070062, −21.65591973916300422170532878130, −20.90382351921010917748897391381, −19.97961404103297597260086587663, −19.31855963869479510539371194151, −18.53378286628732259676348957705, −17.783233684202836364900063779097, −16.498382299316728231625340402046, −15.842934089481936534720936578273, −15.132990128092978934362597078028, −14.45258491679902635497024657940, −13.47216557585835495026428103731, −12.81155916021645605973099868732, −11.74521468286124607995793457234, −10.75405177240989401214631632869, −9.93501923530478267988999851898, −9.07330718043912234495735246006, −8.18348749253856973386536176592, −7.83463169704723502742958536887, −6.13049531855631125885234636757, −5.56964162152594940677871247778, −4.33078594333958220893938520236, −3.29894724819435308824927768291, −2.657220379667556747355065355337, −1.43948151915806662767936372594, 0.869579691472170297448605563014, 2.05422331831308722840356174250, 2.91672772218807014332432575185, 4.183103575743962503345323630, 4.68697100582264074506854765134, 6.450018064162685479556072431200, 7.0914844552354611743815399057, 7.71750183141089928981764243347, 8.84105451413060663947780352005, 9.56812493182015934429433771594, 10.38535752630334179685372742141, 11.53777404616730432656495928067, 12.40709010789690891764426513719, 13.31497437765597812143730314981, 14.00423168576180692322526246078, 14.51260462562445849874050151381, 15.68887068097503394786535833475, 16.28028672387717799044224131329, 17.63108249316599864378208419105, 17.95127542203227275108508208240, 19.11077534975288521414584738024, 19.729578948303058427598273198971, 20.51801260997641467859183460912, 20.93246239191831563486577883585, 22.13141823267987946573813339901

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