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Properties

Label 1-896-896.107-r1-0-0
Degree $1$
Conductor $896$
Sign $0.148 + 0.988i$
Analytic cond. $96.2885$
Root an. cond. $96.2885$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.148 + 0.988i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ 0.148 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.951753341 + 2.540864452i\)
\(L(\frac12)\) \(\approx\) \(2.951753341 + 2.540864452i\)
\(L(1)\) \(\approx\) \(1.713977293 + 0.5692867152i\)
\(L(1)\) \(\approx\) \(1.713977293 + 0.5692867152i\)

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

−21.187635396854399245625705849898, −21.01676240452127302911226053809, −20.0356008940998158519487467673, −19.182587323982665327617912382995, −18.53343025726604868810080385342, −17.6391751547875596625860935656, −16.960230660728131749388151750373, −15.78017136815286926314542574044, −15.218261875659467589157405800903, −14.17511432014348747924462418931, −13.308715088024367920156808147063, −13.17159775437804304065466238687, −12.119278664296521401171079198210, −10.69283524236978626967493855495, −10.20017795674217750569762614427, −8.97242746662445910851019910080, −8.551265071942618505282331260696, −7.76319393834675096007432985928, −6.45036708455399202513054232868, −5.87276712516729391388652649256, −4.67386997411365522960164005702, −3.55308675832371218380198358517, −2.646480782980982315631105473, −1.69755526937270011371893470691, −0.73229186485090750019727582403, 1.2707795348127664743167487249, 2.4363287205662718447623444202, 2.83057118710648165627913019097, 4.26248102000042808844479806917, 4.91529464607484247153643447100, 6.27537754782807659882521229560, 6.977781122749104283967492134654, 8.0273173275118918811833698935, 8.94240887982526309984955435039, 9.60020249458467992010137950660, 10.44008355870992310881237509907, 11.07240629705450083415258846317, 12.54009820413398374771515417670, 13.25895492476366841293500399109, 13.98557590705481369549820871182, 14.60559423030244115475729086332, 15.48846184139602841104825497957, 16.17283925448267128278052986347, 17.23457343732731218509301346251, 18.192178933002755785361840041514, 18.66496990403164169122213427745, 19.62596660198045460402238834757, 20.53399830409626007190018889606, 21.19073889625556633812172117604, 21.55583455666370230624415691312

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