↑

Properties

Label 1-800-800.373-r1-0-0
Degree $1$
Conductor $800$
Sign $-0.701 - 0.712i$
Analytic cond. $85.9719$
Root an. cond. $85.9719$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.156 − 0.987i)3-s + 7-s + (−0.951 + 0.309i)9-s + (0.891 − 0.453i)11-s + (0.453 − 0.891i)13-s + (−0.587 − 0.809i)17-s + (−0.156 + 0.987i)19-s + (−0.156 − 0.987i)21-s + (0.309 − 0.951i)23-s + (0.453 + 0.891i)27-s + (0.987 − 0.156i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.891 − 0.453i)37-s + (−0.951 − 0.309i)39-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)3-s + 7-s + (−0.951 + 0.309i)9-s + (0.891 − 0.453i)11-s + (0.453 − 0.891i)13-s + (−0.587 − 0.809i)17-s + (−0.156 + 0.987i)19-s + (−0.156 − 0.987i)21-s + (0.309 − 0.951i)23-s + (0.453 + 0.891i)27-s + (0.987 − 0.156i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.891 − 0.453i)37-s + (−0.951 − 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ -0.701 - 0.712i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8086519731 - 1.930777958i\)
\(L(\frac12)\) \(\approx\) \(0.8086519731 - 1.930777958i\)
\(L(1)\) \(\approx\) \(1.034078380 - 0.5559566358i\)
\(L(1)\) \(\approx\) \(1.034078380 - 0.5559566358i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−22.04667230759524601876254941186, −21.6521242811852499701613366646, −20.91050653455630049052765693321, −20.0460228722509676013187025382, −19.40067614653091673359904854761, −18.13828908466644350744063138857, −17.29607310269466633340366504524, −16.94900581409592757842047552001, −15.720464824527609232223701058385, −15.1787818414898932233613004938, −14.35568813947170326502478562763, −13.67667514149724526527820338474, −12.34151652119272301346156758836, −11.33632456369380639938698431232, −11.09290264736032497987576068379, −9.9387455306750347504621463868, −9.00090037337994931929131488047, −8.543997663247256104498140130871, −7.189059895802187621519556174245, −6.256798077156142853975660825758, −5.17411512183442536222643567259, −4.36008410042212799369734276113, −3.75460805410632306656386029168, −2.30459007600331450494928644356, −1.210638047292727027278949583491, 0.51741744334426752752913859706, 1.38668284153785974643875175120, 2.39469324081084914412733260866, 3.56561294026189059907370982296, 4.8313652561972862869709240825, 5.74387000811423036894388783737, 6.59968993012330733035742028405, 7.50113848363986511286024296548, 8.39311190259911325621715875885, 8.91422670555284423638684897407, 10.48183136329240413467864803010, 11.151153423200121121190595104302, 11.99234367376281964858514810509, 12.6429305707257862920098172927, 13.77301717649673156436906208287, 14.216964868523150117096732316619, 15.10614037249190148451884491702, 16.322436569568673887896832682498, 17.06729270513051582360182221984, 17.99344191949995363041512227241, 18.29571598554301180734265967285, 19.35428167168075967014911476027, 20.110811940222994962971463518018, 20.82904156467465657992007690578, 21.83684679928582590188244996557

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