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Properties

Label 1-800-800.629-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: $\chi_{800} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (0:\ ),\ -0.331 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5279318094 - 0.7449354096i\)
\(L(\frac12)\) \(\approx\) \(0.5279318094 - 0.7449354096i\)
\(L(1)\) \(\approx\) \(0.7539374220 - 0.2927869470i\)
\(L(1)\) \(\approx\) \(0.7539374220 - 0.2927869470i\)

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

−22.612531122665488343362348400855, −21.60448864778609257904451196299, −21.27475776071164834969761768815, −20.143916487543657554396839078354, −19.30756175251815256294377393992, −18.21861506951178502463793631117, −17.63770409281876040909238357828, −17.0986797060466236666072535697, −15.74891034959308498814999406947, −15.46677979412118070189147674352, −14.75575644045736597016059830279, −13.34266329326651104184909899846, −12.38752897563980931613339050428, −12.04089276669868275273035076653, −10.9307739846948891621898351189, −10.17756669251636438345763642291, −9.41451366278607075855043417324, −8.43337575869009700802013547891, −7.28498369335555412023319808746, −6.36902971844700910864884460611, −5.3423582875734425829334604991, −4.95267755020258788123865547920, −3.6534216664630974253575831599, −2.57089877929035843637937447098, −1.2193647102278179015678039881, 0.54773580249756213204996447375, 1.524110900524368834703853684490, 2.946837142117452838484802801991, 4.17003970807921295089267089222, 5.01239188959979762771775252863, 6.02625695187220645284402392633, 6.869535911940970050690070250202, 7.54478807044398288240494021908, 8.57171160547698379189708420160, 9.83328001696619630321382337347, 10.57006177213013339699075810434, 11.41426870806100928327485580876, 12.01605961325462581079750376572, 13.08855880258771015875902935652, 13.82904598407940585018760864264, 14.42265263961545236316626575359, 16.02704436689718849018928088703, 16.44345425494558599843890712655, 17.0559463385575655323361043547, 18.02830107514704893858629905186, 18.82005360636978312773057638253, 19.34979124558373517139559031990, 20.56606127747037414585296962670, 21.212732184154016016637029267659, 22.208076267817735491470388658168

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