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Properties

Label 1-800-800.317-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  + (−0.453 − 0.891i)3-s + 7-s + (−0.587 + 0.809i)9-s + (−0.156 + 0.987i)11-s + (0.987 − 0.156i)13-s + (0.951 − 0.309i)17-s + (−0.453 + 0.891i)19-s + (−0.453 − 0.891i)21-s + (−0.809 + 0.587i)23-s + (0.987 + 0.156i)27-s + (−0.891 + 0.453i)29-s + (0.309 + 0.951i)31-s + (0.951 − 0.309i)33-s + (0.156 + 0.987i)37-s + (−0.587 − 0.809i)39-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)3-s + 7-s + (−0.587 + 0.809i)9-s + (−0.156 + 0.987i)11-s + (0.987 − 0.156i)13-s + (0.951 − 0.309i)17-s + (−0.453 + 0.891i)19-s + (−0.453 − 0.891i)21-s + (−0.809 + 0.587i)23-s + (0.987 + 0.156i)27-s + (−0.891 + 0.453i)29-s + (0.309 + 0.951i)31-s + (0.951 − 0.309i)33-s + (0.156 + 0.987i)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: $\chi_{800} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (1:\ ),\ -0.331 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5012465172 + 0.7072812681i\)
\(L(\frac12)\) \(\approx\) \(0.5012465172 + 0.7072812681i\)
\(L(1)\) \(\approx\) \(0.9108385591 - 0.05719404173i\)
\(L(1)\) \(\approx\) \(0.9108385591 - 0.05719404173i\)

Euler product

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

−21.62585071230769134080247702466, −21.131151621088075292873489276065, −20.56909503089147935560756382329, −19.45841151359025526998463581359, −18.4264775193667838450853560466, −17.79915382952643529155506324932, −16.793994711193178247043382164061, −16.311831851168090302600579183880, −15.35425291542507199803293780831, −14.62167299588287043993546421318, −13.85346733061549478359995804891, −12.80032839240049944254979996160, −11.51956431962238577384660927790, −11.22397926269091471340599638565, −10.4162753910576644686311978283, −9.38281271109282391268052905595, −8.49404116829390949505371096132, −7.813254731417123063247295736989, −6.22954215914019453388298505031, −5.7514611505349440333586600369, −4.65110765663722756076012177823, −3.92769202293263607587266932562, −2.87053574436634245705274371348, −1.398224339583099140442552495553, −0.204854458893208948390170624221, 1.40621860291298101492467851149, 1.77585937282117850275237608216, 3.24784295355413052664732105988, 4.53893228597987205787388448858, 5.444357137492493347377547265, 6.21137883679708955411039722985, 7.381562112432614679130887640520, 7.88628252796592682606200530039, 8.76507973922200777376215962127, 10.12829617551836292122561363503, 10.858931718434913685477252467988, 11.84824755682035890183369876673, 12.31146691410359832573918998825, 13.3283458073925660239256943768, 14.115779820094467431086843840328, 14.862081920927036778811957822523, 15.941219223659816309962964437794, 16.88741421883156010958874598137, 17.604649731931485016703693994559, 18.32653300844950659152938038059, 18.77829227147491076338926803627, 19.99453090580434616825427089644, 20.64396556244202150166534334715, 21.44545329089473122035916985419, 22.54754586277045082358048528593

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