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Properties

Label 1-896-896.523-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  + (−0.321 + 0.946i)3-s + (0.442 − 0.896i)5-s + (−0.793 − 0.608i)9-s + (−0.751 − 0.659i)11-s + (−0.555 + 0.831i)13-s + (0.707 + 0.707i)15-s + (0.258 + 0.965i)17-s + (0.0654 − 0.997i)19-s + (−0.608 + 0.793i)23-s + (−0.608 − 0.793i)25-s + (0.831 − 0.555i)27-s + (−0.195 − 0.980i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s + (0.896 + 0.442i)37-s + ⋯
L(s)  = 1  + (−0.321 + 0.946i)3-s + (0.442 − 0.896i)5-s + (−0.793 − 0.608i)9-s + (−0.751 − 0.659i)11-s + (−0.555 + 0.831i)13-s + (0.707 + 0.707i)15-s + (0.258 + 0.965i)17-s + (0.0654 − 0.997i)19-s + (−0.608 + 0.793i)23-s + (−0.608 − 0.793i)25-s + (0.831 − 0.555i)27-s + (−0.195 − 0.980i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s + (0.896 + 0.442i)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: $\chi_{896} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.759 - 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08280926795 - 0.2241616601i\)
\(L(\frac12)\) \(\approx\) \(0.08280926795 - 0.2241616601i\)
\(L(1)\) \(\approx\) \(0.7327334692 + 0.04533032558i\)
\(L(1)\) \(\approx\) \(0.7327334692 + 0.04533032558i\)

Euler product

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

−22.544103069208640045696620248287, −21.63719611360501407218415302809, −20.5083334822837772777751795936, −19.89979398803418818361338620894, −18.79744558477339973669135639792, −18.16554690775483807761088631683, −17.91089935528763285712106773245, −16.83352705844536692176740389683, −15.98678454014906664638821797240, −14.73706661893491161116024081759, −14.33489237190605597139421371222, −13.33126623631481723668167259564, −12.581345106363411447128698305, −11.91885696316157902705709482082, −10.746591821910975095032401195077, −10.30641937667341118903548321049, −9.227142018820607234397876579493, −7.77947844604770520049295920813, −7.51539821671260192444189033649, −6.513094506012076573223893404386, −5.672579561808372965628358241277, −4.90401731896585880048691946947, −3.23175839523261236222837547535, −2.49370495166856017922424735366, −1.57993445097750236700141063216, 0.10232571334236938463080712563, 1.67365918817277059367224118994, 2.90487869943404345107294777355, 4.0740741132298325849682924877, 4.8058277659087019302772855208, 5.632247505748311732115255693153, 6.32455508259618210444726092657, 7.82358537287316100676679992035, 8.63441750683931199313837784223, 9.54320672807983758012274409316, 9.99164450716330203962360708375, 11.16869015516492796865778829541, 11.71929968729607697986540802028, 12.860089656815081513415522935, 13.53719738631228870315616464024, 14.53638839803789421837800254013, 15.405233965786670290907512418543, 16.18195080432948953150071104338, 16.82320404083666843121630146880, 17.39239521011964911229426278567, 18.3499049784827273829788158018, 19.515397856892470095593970297922, 20.15193907574178927993593198715, 21.11733648729703405168660715615, 21.61258201433754648754778224511

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