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Properties

Label 1-640-640.163-r0-0-0
Degree $1$
Conductor $640$
Sign $-0.918 + 0.395i$
Analytic cond. $2.97214$
Root an. cond. $2.97214$
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  + (0.195 − 0.980i)3-s + (−0.382 − 0.923i)7-s + (−0.923 − 0.382i)9-s + (−0.195 − 0.980i)11-s + (−0.831 + 0.555i)13-s + (−0.707 + 0.707i)17-s + (0.831 − 0.555i)19-s + (−0.980 + 0.195i)21-s + (−0.923 − 0.382i)23-s + (−0.555 + 0.831i)27-s + (0.195 − 0.980i)29-s + i·31-s − 33-s + (−0.555 + 0.831i)37-s + (0.382 + 0.923i)39-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)3-s + (−0.382 − 0.923i)7-s + (−0.923 − 0.382i)9-s + (−0.195 − 0.980i)11-s + (−0.831 + 0.555i)13-s + (−0.707 + 0.707i)17-s + (0.831 − 0.555i)19-s + (−0.980 + 0.195i)21-s + (−0.923 − 0.382i)23-s + (−0.555 + 0.831i)27-s + (0.195 − 0.980i)29-s + i·31-s − 33-s + (−0.555 + 0.831i)37-s + (0.382 + 0.923i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.918 + 0.395i$
Analytic conductor: \(2.97214\)
Root analytic conductor: \(2.97214\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 640,\ (0:\ ),\ -0.918 + 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1138928745 - 0.5528835604i\)
\(L(\frac12)\) \(\approx\) \(-0.1138928745 - 0.5528835604i\)
\(L(1)\) \(\approx\) \(0.6845943742 - 0.4325998968i\)
\(L(1)\) \(\approx\) \(0.6845943742 - 0.4325998968i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.195 - 0.980i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (-0.195 - 0.980i)T \)
13 \( 1 + (-0.831 + 0.555i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (0.831 - 0.555i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (0.195 - 0.980i)T \)
31 \( 1 + iT \)
37 \( 1 + (-0.555 + 0.831i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (-0.195 - 0.980i)T \)
47 \( 1 + (-0.707 + 0.707i)T \)
53 \( 1 + (0.980 - 0.195i)T \)
59 \( 1 + (0.555 - 0.831i)T \)
61 \( 1 + (-0.980 - 0.195i)T \)
67 \( 1 + (0.195 - 0.980i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.555 + 0.831i)T \)
89 \( 1 + (-0.382 - 0.923i)T \)
97 \( 1 - T \)
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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

−22.98287937874552864276966255214, −22.421449022265984470661051579662, −21.87963140884548541199050917104, −20.89366399436230083228488810619, −20.120221895030119245776626005975, −19.5621037179956532037877243764, −18.254928510220038447012598228338, −17.696419547008013939308475013501, −16.51734166522015906954650457070, −15.81192773402012346717200521211, −15.15548975056028067681659680536, −14.46773842402439607666676064166, −13.38309707306827169697520950672, −12.271949158349656594184047832183, −11.675013403940585065557130306921, −10.41786733161578038766176452702, −9.72842917773705147851559722382, −9.11854061484856258808039190635, −8.0457964359838637810245480448, −7.046888027229182967471197493828, −5.64981205032613686618302184129, −5.10379086241672979063401363792, −4.015417116020259586723778520541, −2.90158699734350749673325981112, −2.124088352879138269277734587942, 0.25572888522034828130342025589, 1.54875314137380957448917017656, 2.723513009068045959721359308634, 3.678503262872288429082680864105, 4.92023696279692188116009033296, 6.23478889579789948949516292866, 6.80430520107412843788022273915, 7.76847506750906865294025503153, 8.53694880994939129077276229688, 9.612339183576832108449209637356, 10.63679954591351553744395230720, 11.593116692353837095539977051464, 12.37555959039477500464873525946, 13.53680068177872414579534265979, 13.701308978996166819788484467489, 14.72551375154077931459866407970, 15.917699857946659301884666196037, 16.80844890922751870842573943093, 17.4999892699305194951378615341, 18.3702716155967914676067997018, 19.373063641246697529923975278931, 19.70027688916592468000493393693, 20.591462759034462115615618185652, 21.750296860893450043415830223524, 22.49076349442244588241921181084

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