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Properties

Label 1-1455-1455.158-r0-0-0
Degree $1$
Conductor $1455$
Sign $-0.835 - 0.550i$
Analytic cond. $6.75699$
Root an. cond. $6.75699$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s − i·8-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − 19-s − i·22-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + i·28-s + (−0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)7-s − i·8-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − 19-s − i·22-s + (0.866 + 0.5i)23-s + (0.5 − 0.866i)26-s + i·28-s + (−0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1455 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1455\)    =    \(3 \cdot 5 \cdot 97\)
Sign: $-0.835 - 0.550i$
Analytic conductor: \(6.75699\)
Root analytic conductor: \(6.75699\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1455} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1455,\ (0:\ ),\ -0.835 - 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5380905472 - 1.794631738i\)
\(L(\frac12)\) \(\approx\) \(0.5380905472 - 1.794631738i\)
\(L(1)\) \(\approx\) \(1.258073037 - 0.7632706124i\)
\(L(1)\) \(\approx\) \(1.258073037 - 0.7632706124i\)

Euler product

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

−21.15118008028707649115690877812, −20.320636121241749524480890699371, −19.71710102864112248280373713085, −18.85283471239855547913223176696, −17.6778971592092250409252533878, −17.12910481529445568244612436648, −16.31820081981094542820392484791, −15.749491273900220135873081648468, −14.86445159174952450853818914643, −14.27018309296754812244534067603, −13.30347600763356347385753301595, −12.81241085707563821476361376772, −12.17161087693306573480186509374, −10.98881379646705375036718981305, −10.542162232710008643098845421633, −9.10379532825460957665212156948, −8.69265191940077030782902598395, −7.34272793219696082361611993400, −6.77007041668284652591850978917, −6.25777560041449916504737916329, −5.14554592590949956684364459505, −4.11207757655898962912372560905, −3.769136087323160002990982592443, −2.579160739411999704304299962787, −1.56743030889104601687067875313, 0.48977312717600109196193159733, 1.76999353103701326850076762639, 2.80727498653500068316007453639, 3.46694564607871680979365731263, 4.29476037928873399318971492160, 5.38244301368508398438399220539, 6.22221004547505511475018056961, 6.55172116537211606724889102114, 7.887498517842419239841765346475, 9.10289068658851419329495987967, 9.468473695037928209850124644049, 10.847083931336196345514260889712, 11.07155402957749084446080920579, 12.075027328441040265912487481103, 12.92521471388295582673186734176, 13.3786296551688509903018575954, 14.12253910888629406183264844038, 15.25170522241723952666929288712, 15.557785222015823929287237048774, 16.426494104466886314848711620373, 17.31723734969849080671804023915, 18.565398339123499317304991161591, 19.004349019984289904474714541301, 19.67114155130038157239792114304, 20.51804890640958928487221213259

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