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Properties

Label 1-896-896.451-r0-0-0
Degree $1$
Conductor $896$
Sign $-0.759 + 0.650i$
Analytic cond. $4.16100$
Root an. cond. $4.16100$
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  + (−0.946 + 0.321i)3-s + (−0.896 + 0.442i)5-s + (0.793 − 0.608i)9-s + (−0.659 − 0.751i)11-s + (−0.831 + 0.555i)13-s + (0.707 − 0.707i)15-s + (0.258 − 0.965i)17-s + (0.997 − 0.0654i)19-s + (0.608 + 0.793i)23-s + (0.608 − 0.793i)25-s + (−0.555 + 0.831i)27-s + (0.980 + 0.195i)29-s + (−0.866 + 0.5i)31-s + (0.866 + 0.5i)33-s + (0.442 + 0.896i)37-s + ⋯
L(s)  = 1  + (−0.946 + 0.321i)3-s + (−0.896 + 0.442i)5-s + (0.793 − 0.608i)9-s + (−0.659 − 0.751i)11-s + (−0.831 + 0.555i)13-s + (0.707 − 0.707i)15-s + (0.258 − 0.965i)17-s + (0.997 − 0.0654i)19-s + (0.608 + 0.793i)23-s + (0.608 − 0.793i)25-s + (−0.555 + 0.831i)27-s + (0.980 + 0.195i)29-s + (−0.866 + 0.5i)31-s + (0.866 + 0.5i)33-s + (0.442 + 0.896i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.759 + 0.650i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.759 + 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1274332391 + 0.3449571183i\)
\(L(\frac12)\) \(\approx\) \(0.1274332391 + 0.3449571183i\)
\(L(1)\) \(\approx\) \(0.5675689319 + 0.1216583152i\)
\(L(1)\) \(\approx\) \(0.5675689319 + 0.1216583152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.946 + 0.321i)T \)
5 \( 1 + (-0.896 + 0.442i)T \)
11 \( 1 + (-0.659 - 0.751i)T \)
13 \( 1 + (-0.831 + 0.555i)T \)
17 \( 1 + (0.258 - 0.965i)T \)
19 \( 1 + (0.997 - 0.0654i)T \)
23 \( 1 + (0.608 + 0.793i)T \)
29 \( 1 + (0.980 + 0.195i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (0.442 + 0.896i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (-0.195 - 0.980i)T \)
47 \( 1 + (-0.965 + 0.258i)T \)
53 \( 1 + (-0.659 - 0.751i)T \)
59 \( 1 + (0.0654 - 0.997i)T \)
61 \( 1 + (0.751 + 0.659i)T \)
67 \( 1 + (-0.946 + 0.321i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (0.130 + 0.991i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 + (0.555 + 0.831i)T \)
89 \( 1 + (-0.991 - 0.130i)T \)
97 \( 1 - iT \)
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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

−21.78033336451004851114230124341, −20.83642677438022826670456031283, −19.97154277508475452169584885204, −19.28629147406005285474774399004, −18.38114131321137338317025679641, −17.67489300133618802422957725703, −16.862263005998378084685248970800, −16.17190188171875038583433944386, −15.378086763371878291586722242693, −14.647295854687176654054093144374, −13.23663288106752620340102172329, −12.51684279040011468582734323851, −12.16970317337154958075417570812, −11.12262066333800344067585170357, −10.40836711658911092288945610442, −9.51183442103242521053799754528, −8.14388497035535092574205471387, −7.590937857051527682384519180378, −6.80125707732614327005867981089, −5.56457467127418827654581842801, −4.92285948667712200499604937127, −4.09541787334522033348487849823, −2.76543984132870698903313208802, −1.44948459028528043252857225204, −0.22710771092963883317627859895, 1.07851826766418206022420457486, 2.844858944278986562613072429558, 3.589798955505522716797805483592, 4.852207019055314585877523125787, 5.27501256287665996656778535162, 6.61272305567970618551517089873, 7.2194112975902621678513421706, 8.11527227411929503332310047059, 9.38387902261970562868543710472, 10.129468990491094600816943807088, 11.13502413974128244396836183193, 11.59475323359171904273748374815, 12.27343450317303361019568422265, 13.37183274852629884627126870706, 14.37408500159725561422026692388, 15.25130164864779525759150586552, 16.12614656205514854677460721581, 16.38130318820135124065612740275, 17.53820547122339882972732003879, 18.34520883969958461062898315824, 18.92757069100868584039007890037, 19.84017290149024463837114769715, 20.78926167206201661314431240329, 21.71809934818192865607502655564, 22.21021849201537259533815997242

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