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Properties

Label 1-555-555.167-r1-0-0
Degree $1$
Conductor $555$
Sign $0.775 - 0.630i$
Analytic cond. $59.6430$
Root an. cond. $59.6430$
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  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.642 + 0.766i)7-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.866 + 0.5i)14-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + (−0.984 − 0.173i)19-s + (−0.766 + 0.642i)22-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)26-s + (0.342 − 0.939i)28-s + (−0.866 + 0.5i)29-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.642 + 0.766i)7-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.866 + 0.5i)14-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + (−0.984 − 0.173i)19-s + (−0.766 + 0.642i)22-s + (0.5 − 0.866i)23-s + (0.5 + 0.866i)26-s + (0.342 − 0.939i)28-s + (−0.866 + 0.5i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.775 - 0.630i$
Analytic conductor: \(59.6430\)
Root analytic conductor: \(59.6430\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (1:\ ),\ 0.775 - 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7272335213 - 0.2582930776i\)
\(L(\frac12)\) \(\approx\) \(0.7272335213 - 0.2582930776i\)
\(L(1)\) \(\approx\) \(0.6204502527 - 0.2562913442i\)
\(L(1)\) \(\approx\) \(0.6204502527 - 0.2562913442i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.642 + 0.766i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.642 - 0.766i)T \)
83 \( 1 + (0.342 - 0.939i)T \)
89 \( 1 + (-0.642 - 0.766i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−23.38317251808052255150185717008, −22.59043090285225904476237007853, −21.882146281430774896593195950687, −20.62536649088930106198540604417, −19.62804521422139154285255668046, −19.08776329606792711956092931596, −17.81417255344885897144721901338, −17.30638182316751794738958397290, −16.572434099595292035271403526113, −15.4595797538349213135336952204, −15.0542090998898946807238785558, −13.9296730562130393595379268060, −13.0684744303095321140060780779, −12.49881806894415294949675875681, −10.82488734619368952664818196811, −10.05594530031297793390247367349, −9.31192398368399081590663298411, −8.1922062464513762239492938537, −7.22362268698080149692203344942, −6.738538406903462590986573710895, −5.5004596203948550967898758595, −4.58055667653462098340103107288, −3.64924574193172831220930458771, −2.05868752700066672825132135249, −0.40449164849221201346190762549, 0.50981058885738469159092059778, 2.25816804077885168991046136174, 2.73273471329367631089251143348, 4.00947055391884852098253695452, 5.0241843398549071723116373932, 6.07969523023551621170776263547, 7.33747344041240071541196503753, 8.65241617766871565079729502450, 9.07642997409647259470148858455, 10.137741665266616571424925779235, 11.00604368470899753186624665331, 11.81724917417407239022991582573, 12.7880272697115716752256486219, 13.2766752524229449457552442072, 14.430016250744098130211100240583, 15.36150986681429991957605223788, 16.50700505240907234903713039966, 17.214521103774986405335848392457, 18.40599488132985686967123805485, 18.88310434425377045138864239894, 19.60137159630583827304415014781, 20.528502486580664740822155972917, 21.437059783094731677866921341490, 22.07890432530766922511372625574, 22.6447134103529214572094688896

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