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Properties

Label 1-200-200.77-r1-0-0
Degree $1$
Conductor $200$
Sign $-0.684 + 0.728i$
Analytic cond. $21.4929$
Root an. cond. $21.4929$
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  + (0.587 − 0.809i)3-s + i·7-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)21-s + (−0.951 − 0.309i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (−0.809 + 0.587i)31-s + (0.587 + 0.809i)33-s + (0.951 − 0.309i)37-s + (−0.309 + 0.951i)39-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)3-s + i·7-s + (−0.309 − 0.951i)9-s + (−0.309 + 0.951i)11-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.809 + 0.587i)21-s + (−0.951 − 0.309i)23-s + (−0.951 − 0.309i)27-s + (−0.809 − 0.587i)29-s + (−0.809 + 0.587i)31-s + (0.587 + 0.809i)33-s + (0.951 − 0.309i)37-s + (−0.309 + 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(200\)    =    \(2^{3} \cdot 5^{2}\)
Sign: $-0.684 + 0.728i$
Analytic conductor: \(21.4929\)
Root analytic conductor: \(21.4929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{200} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 200,\ (1:\ ),\ -0.684 + 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1808222183 + 0.4178554921i\)
\(L(\frac12)\) \(\approx\) \(0.1808222183 + 0.4178554921i\)
\(L(1)\) \(\approx\) \(0.9106788287 - 0.04763445966i\)
\(L(1)\) \(\approx\) \(0.9106788287 - 0.04763445966i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.951 - 0.309i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.951 - 0.309i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−26.3399567132275247906684605691, −25.715675976521947395298814056291, −24.3302557995576768533748384505, −23.64984144075012662066933372276, −22.154292080258532003752533483079, −21.726355178722290213733349937962, −20.46310423973642726777600691482, −19.850270084358881422157870509599, −18.944184389639793188358236984585, −17.36661020142745531269642839289, −16.67346426745383522153912118791, −15.615544586626798982808133886198, −14.66505804627662494543007036844, −13.751771545592860299713606840170, −12.84723939765704574500519476534, −11.09641006636492316963449773345, −10.50829856762303070631454064442, −9.407263978368161690313007448253, −8.30234042680387034893797904134, −7.33850640497051947176237295480, −5.764186606332386498080931118434, −4.45164724144991272888725355736, −3.56759334461031870579497447544, −2.214227213722598517530482874205, −0.12915188341197607412352431004, 1.95261820085388934466319424543, 2.61185764707495813450639212093, 4.34059847224317541790717918597, 5.7693272196321294734715896943, 6.95215786765878119295838077778, 7.90185327690101861268507584968, 9.01354178901392996577579008560, 9.88420115152232030568452172962, 11.61797539310150528580744934957, 12.3979918144459498444377864619, 13.19895138659645850232766226942, 14.58114795575136720959541708625, 15.03990670173766019574308212198, 16.39491675508997937672643613049, 17.81681011429909814931105087420, 18.3337765693864803359123614644, 19.39069241345832200350287736401, 20.19937724624678096918584371762, 21.23350642811612277488993742967, 22.334680211343152023119107185870, 23.35732405102949668102964038331, 24.4285744443214487892426826947, 25.05269440651141565941747406070, 25.87050739845374698531922050182, 26.862411118094835561518356596015

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