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Properties

Label 1-555-555.308-r0-0-0
Degree $1$
Conductor $555$
Sign $0.814 - 0.580i$
Analytic cond. $2.57740$
Root an. cond. $2.57740$
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  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−0.642 − 0.766i)7-s + (0.866 + 0.5i)8-s + (0.5 − 0.866i)11-s + (0.342 − 0.939i)13-s + (−0.5 − 0.866i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (−0.173 − 0.984i)19-s + (0.642 − 0.766i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + (−0.342 − 0.939i)28-s + (−0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.939 + 0.342i)4-s + (−0.642 − 0.766i)7-s + (0.866 + 0.5i)8-s + (0.5 − 0.866i)11-s + (0.342 − 0.939i)13-s + (−0.5 − 0.866i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (−0.173 − 0.984i)19-s + (0.642 − 0.766i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + (−0.342 − 0.939i)28-s + (−0.5 + 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.814 - 0.580i$
Analytic conductor: \(2.57740\)
Root analytic conductor: \(2.57740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (308, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 555,\ (0:\ ),\ 0.814 - 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.355318078 - 0.7539553435i\)
\(L(\frac12)\) \(\approx\) \(2.355318078 - 0.7539553435i\)
\(L(1)\) \(\approx\) \(1.834771868 - 0.1843964985i\)
\(L(1)\) \(\approx\) \(1.834771868 - 0.1843964985i\)

Euler product

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

−23.23240241926404223560232361262, −22.544976025569680975739753840356, −21.881174093751746219389349895880, −21.0528743653462022021882974848, −20.260722946661164241012556903985, −19.26304018966877506252445169635, −18.76973996656546585295849872277, −17.38757002525853640527731671974, −16.43771108728844253734559682868, −15.63236515036870481931175845436, −14.882235786255745018625700183062, −14.083650450210281792257428781059, −13.11240103157767658946223919889, −12.22025268844630488733041703753, −11.83269034862214344587705760713, −10.548543605578347626157282140270, −9.73612220997768761109156514665, −8.65408806809614731886538364193, −7.36097083967403906841421098630, −6.27768492363226211763299947733, −5.88119524717825715035856697372, −4.37976499748835482860667170240, −3.85328165749643857037475720930, −2.455833226082390373413536945126, −1.702391236143539635179291508611, 0.96955339649217771792661216263, 2.67024316211382509900490629141, 3.45573329329340862375225892649, 4.37279686711893317550176754434, 5.50797717676658209745878368231, 6.39266317623884729701562846258, 7.18752686403564802790793380769, 8.17537363719865106228367652804, 9.41483587636377075560385570726, 10.60375105202814333775104332163, 11.271810059038740418692510280327, 12.25528927177689873707811913750, 13.38675913342558181621362760954, 13.59688296466967748042754939502, 14.66518671191919367535189663486, 15.730084755781279920847567217485, 16.23689224395595506896109560937, 17.14109122482896206214589685006, 18.10232534302123484522379052058, 19.55589730805224222396501380912, 19.90638505463665359543136736204, 20.8410146175543048516827839150, 21.79792994403591922644932729011, 22.51441062109322313364034564364, 23.1107844433822600765271580665

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