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Properties

Label 1-800-800.19-r1-0-0
Degree $1$
Conductor $800$
Sign $-0.331 + 0.943i$
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

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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

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

Invariants

Degree: \(1\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $-0.331 + 0.943i$
Analytic conductor: \(85.9719\)
Root analytic conductor: \(85.9719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ -0.331 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5253592771 + 0.7413054514i\)
\(L(\frac12)\) \(\approx\) \(0.5253592771 + 0.7413054514i\)
\(L(1)\) \(\approx\) \(0.7585730612 + 0.1298165265i\)
\(L(1)\) \(\approx\) \(0.7585730612 + 0.1298165265i\)

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

−21.92285016009077671222496646185, −21.265839538973458707592318301844, −20.19767000830542718060563374096, −19.18261657383008883144802041222, −18.30977979041237281235088142514, −18.162959865970230592423158676135, −16.97089789717461732344659136778, −16.154763312330656098974855039214, −15.62998208286061716405727651691, −14.47389902020056063455246558284, −13.472263066318251657629016988329, −12.77134836272798039646098662452, −11.821876775349188193207760312236, −11.397383657526372587300179949797, −10.33697090989165976251461473693, −9.46253903640168028109457064823, −8.243639197989534263818321964264, −7.64204628871880137263030648390, −6.42221956375525814735605543137, −5.52715279381647065150653821537, −5.29404845714990935333061664304, −3.65143899243899788453249435375, −2.61720214183429928002763676070, −1.39156598643329449301295984074, −0.29068264961153479223855864911, 0.909511316323030823552923396889, 2.069967929046212560613599249641, 3.76552823244260322383339523542, 4.309923192333577586643339013682, 5.19749966372770112087948097757, 6.36968834895304663837838056166, 6.99859506857978017658289565817, 7.95615389161246388611758769348, 9.395370644662646373205117704599, 9.896215280222446667199839422583, 10.89297355453597363285815734376, 11.40576188600465031549969274464, 12.51999190781542276968495514613, 13.13834730550588563995598430540, 14.35100763187869446755125659257, 15.018228036534397167722716107945, 16.109189730215973114639104338032, 16.64390509259259370987738691549, 17.4238256851642772013688232086, 18.059964885601092691040143861074, 19.05579416312523889257206465328, 20.126457055125500831681423479998, 20.70978888066863754358732297636, 21.59351826836853530782768870381, 22.388526624255624534279190273372

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