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Properties

Label 1-555-555.113-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: $\chi_{555} (113, \cdot )$
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

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

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