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Properties

Label 1-800-800.29-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  + (0.453 − 0.891i)3-s − i·7-s + (−0.587 − 0.809i)9-s + (−0.987 + 0.156i)11-s + (−0.987 − 0.156i)13-s + (0.309 − 0.951i)17-s + (−0.891 + 0.453i)19-s + (0.891 + 0.453i)21-s + (−0.587 + 0.809i)23-s + (−0.987 + 0.156i)27-s + (−0.453 + 0.891i)29-s + (0.309 − 0.951i)31-s + (−0.309 + 0.951i)33-s + (−0.156 + 0.987i)37-s + (−0.587 + 0.809i)39-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)3-s − i·7-s + (−0.587 − 0.809i)9-s + (−0.987 + 0.156i)11-s + (−0.987 − 0.156i)13-s + (0.309 − 0.951i)17-s + (−0.891 + 0.453i)19-s + (0.891 + 0.453i)21-s + (−0.587 + 0.809i)23-s + (−0.987 + 0.156i)27-s + (−0.453 + 0.891i)29-s + (0.309 − 0.951i)31-s + (−0.309 + 0.951i)33-s + (−0.156 + 0.987i)37-s + (−0.587 + 0.809i)39-s + ⋯

Functional equation

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

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: $\chi_{800} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 800,\ (0:\ ),\ -0.744 + 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04250856412 + 0.1111733464i\)
\(L(\frac12)\) \(\approx\) \(0.04250856412 + 0.1111733464i\)
\(L(1)\) \(\approx\) \(0.8012595597 - 0.1367036863i\)
\(L(1)\) \(\approx\) \(0.8012595597 - 0.1367036863i\)

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

−21.71331144275658151839341429181, −21.23827094796171116922119523525, −20.3778408448262134816532498905, −19.676836933052832004136153616146, −19.037847695193154673421330969675, −17.77299379014134676394262435372, −16.89062606224315196531493894315, −16.43866376682529345808849511912, −15.34756011111836928847672829910, −14.774973991902258944231016955943, −13.88479267140150796807651971847, −13.16709709539713813439354494066, −12.144089677758760625837874576609, −10.82675901547638802747535171361, −10.42488257574683077965968395173, −9.720759433706399579253851219146, −8.540253288905647716189132671046, −7.90675239673290453980509324982, −6.917425690658116077655751551943, −5.66984405148477398509807121441, −4.65723960265707927900875677508, −4.03391432838767572045172990684, −2.95106820804787292862133007563, −1.98565617402799615744960376189, −0.04555301495920289678951418276, 1.73494179078893221970259443986, 2.51024595173160923429923743381, 3.29434894729724190574792750175, 4.88537996274905650143417282645, 5.67379227436974365724388211081, 6.65838029670309328386975771197, 7.68417758794304441638442131427, 8.18372631313619838691075956705, 9.26845596749209242496965962810, 9.96872500358544843809449821995, 11.31001045610457640719648233338, 12.142710594917509319288117147298, 12.7055670708916418888306717265, 13.53024323680997959519023818268, 14.50642009757676670695466342066, 15.14088748874063447155122326396, 15.97916994312135282070173120546, 17.16432044842582901446329578993, 17.920912551520060547818648430639, 18.7372392084009405479349240200, 19.080980772084968858861119701865, 20.21554841495397589858893030492, 20.80338483192481706428423355637, 21.79117766565311333907165157615, 22.586526887724518826078198687921

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