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Properties

Label 1-728-728.627-r1-0-0
Degree $1$
Conductor $728$
Sign $0.949 - 0.313i$
Analytic cond. $78.2344$
Root an. cond. $78.2344$
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  + 3-s + (0.5 − 0.866i)5-s + 9-s + 11-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + 19-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + 33-s + (0.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + 3-s + (0.5 − 0.866i)5-s + 9-s + 11-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)17-s + 19-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 27-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + 33-s + (0.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.949 - 0.313i$
Analytic conductor: \(78.2344\)
Root analytic conductor: \(78.2344\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 728,\ (1:\ ),\ 0.949 - 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.132345428 - 0.6642529918i\)
\(L(\frac12)\) \(\approx\) \(4.132345428 - 0.6642529918i\)
\(L(1)\) \(\approx\) \(1.895518411 - 0.2008129936i\)
\(L(1)\) \(\approx\) \(1.895518411 - 0.2008129936i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 + T \)
5 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−22.29054443068276131506515810303, −21.5996038260292367818951668415, −20.64785344708540052096257130662, −19.99056503030891244246917512134, −19.135296837828864162646719156769, −18.39813512901564611799292693184, −17.73737393237501344792804530933, −16.598510123727063453799459328391, −15.64937374968051452466225738874, −14.795202990683810995534331573089, −14.14090959391754946935549756168, −13.637087599231907276512552220959, −12.57290759205840898212248227452, −11.49693980345364036047443486610, −10.5897055922249629831397635472, −9.54769348370674301662890789002, −9.15549986306049931196503371137, −7.96406598803877596587686421922, −7.00828189134731875011306096074, −6.453297301305768878679401004571, −5.050807308491987311024771749266, −3.91319835375859133969876751498, −2.99810407164146274450221646152, −2.234811168482889098095890390251, −1.037949062367644411465999224478, 1.073761128736046900062890964389, 1.757269777851103122339865911511, 3.00908017389289188865414666634, 4.02653210235416116933276302246, 4.86935570857464899319884692278, 6.06944222243564519116888097102, 7.0468174631225349582505778965, 8.15444222517794560615608391887, 8.83447631354467650587895888680, 9.535245478898973764357925547597, 10.279332324869489432579029960799, 11.70068388754443883784275281309, 12.441430971130005835163371146549, 13.60076420591438629906994450092, 13.70921917518267433404055628950, 14.95663034169509286613975279628, 15.58535702924486723302017801597, 16.643665860659053857046184896834, 17.342214674499793556008204055, 18.25781992986216835878091205932, 19.3386741327195409785050178888, 19.93020992448212295485726825307, 20.50695797975080073600860404412, 21.56355974317543344113636608530, 21.85690148475645920907005172458

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