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Properties

Label 1-800-800.403-r0-0-0
Degree $1$
Conductor $800$
Sign $0.744 - 0.667i$
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 + 7-s + (0.587 + 0.809i)9-s + (−0.987 + 0.156i)11-s + (0.156 − 0.987i)13-s + (0.951 + 0.309i)17-s + (0.891 − 0.453i)19-s + (−0.891 − 0.453i)21-s + (−0.809 − 0.587i)23-s + (−0.156 − 0.987i)27-s + (−0.453 + 0.891i)29-s + (−0.309 + 0.951i)31-s + (0.951 + 0.309i)33-s + (0.987 + 0.156i)37-s + (−0.587 + 0.809i)39-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)3-s + 7-s + (0.587 + 0.809i)9-s + (−0.987 + 0.156i)11-s + (0.156 − 0.987i)13-s + (0.951 + 0.309i)17-s + (0.891 − 0.453i)19-s + (−0.891 − 0.453i)21-s + (−0.809 − 0.587i)23-s + (−0.156 − 0.987i)27-s + (−0.453 + 0.891i)29-s + (−0.309 + 0.951i)31-s + (0.951 + 0.309i)33-s + (0.987 + 0.156i)37-s + (−0.587 + 0.809i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.744 - 0.667i$
Analytic conductor: \(3.71518\)
Root analytic conductor: \(3.71518\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (0:\ ),\ 0.744 - 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080968930 - 0.4133224243i\)
\(L(\frac12)\) \(\approx\) \(1.080968930 - 0.4133224243i\)
\(L(1)\) \(\approx\) \(0.9037210673 - 0.1647684250i\)
\(L(1)\) \(\approx\) \(0.9037210673 - 0.1647684250i\)

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 \( 1 + T \)
11 \( 1 + (-0.987 + 0.156i)T \)
13 \( 1 + (0.156 - 0.987i)T \)
17 \( 1 + (0.951 + 0.309i)T \)
19 \( 1 + (0.891 - 0.453i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.453 + 0.891i)T \)
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 \( 1 + (0.951 - 0.309i)T \)
53 \( 1 + (0.453 - 0.891i)T \)
59 \( 1 + (-0.156 + 0.987i)T \)
61 \( 1 + (0.156 + 0.987i)T \)
67 \( 1 + (-0.453 - 0.891i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.453 - 0.891i)T \)
89 \( 1 + (0.587 - 0.809i)T \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−22.26479021666622877157585511808, −21.52668822111900672538076494507, −20.895868975736472239421901926778, −20.318102351678765033814673916706, −18.67099379166396609724469745275, −18.47061649538249060089407744682, −17.46869051585813927091748264379, −16.70778231317225571435991555303, −16.007727643762523091210214368899, −15.21141800762414300145286240574, −14.26403974897847873188433258938, −13.46801573318234425934483874468, −12.23238772658128243594939441510, −11.608588846831830956185358171297, −10.98325221599886788189514035587, −9.99641368775566094881090593005, −9.32254237903601287332960515631, −7.95429524342163516388077478845, −7.401540397102884328550075152916, −5.98771876891893926085326131486, −5.44594216055480581731559163588, −4.51691046215438179483130972866, −3.66185776803618705272770348333, −2.18097192878865124196736999556, −0.99035725275496042617260735029, 0.79348077733514253460288795296, 1.8415118611705322194873876625, 3.03747983764249007481193137279, 4.47867143667788716246187979880, 5.37167558944257799705450466241, 5.79701002356323300937588564469, 7.26906580103281203361988306774, 7.72008481914626513683507583669, 8.63324153372599087678143793082, 10.18686356785042841652498000498, 10.59441945778950192495837351615, 11.529630824558390989807319891413, 12.32637299145347527258309827921, 13.03149931053138310251778574435, 13.96379200600026157211204041054, 14.90308136316050986379038144188, 15.85673409085428454314257796696, 16.56495830982116129951514108574, 17.58153439191586253605660070277, 18.12737493819899537475086048761, 18.565612627980538085014264754588, 19.84059537509283258898908479274, 20.622592177374659739882207087347, 21.448119086482949415428622312, 22.24350650946235957300689580267

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