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Properties

Label 1-896-896.53-r0-0-0
Degree $1$
Conductor $896$
Sign $-0.244 + 0.969i$
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  + (0.659 + 0.751i)3-s + (−0.997 + 0.0654i)5-s + (−0.130 + 0.991i)9-s + (0.946 − 0.321i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)15-s + (0.965 + 0.258i)17-s + (0.896 − 0.442i)19-s + (−0.991 − 0.130i)23-s + (0.991 − 0.130i)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 + (0.0654 + 0.997i)37-s + ⋯
L(s)  = 1  + (0.659 + 0.751i)3-s + (−0.997 + 0.0654i)5-s + (−0.130 + 0.991i)9-s + (0.946 − 0.321i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)15-s + (0.965 + 0.258i)17-s + (0.896 − 0.442i)19-s + (−0.991 − 0.130i)23-s + (0.991 − 0.130i)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 + (0.0654 + 0.997i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.244 + 0.969i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.244 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8987445005 + 1.154100975i\)
\(L(\frac12)\) \(\approx\) \(0.8987445005 + 1.154100975i\)
\(L(1)\) \(\approx\) \(1.049773562 + 0.4578821821i\)
\(L(1)\) \(\approx\) \(1.049773562 + 0.4578821821i\)

Euler product

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

−21.73373825950964222048071533013, −20.55840546678351420440576538862, −20.02404899596326292816045237785, −19.4795196100312820747376448592, −18.694036223665305952738104270871, −17.90847710738900803378083188708, −17.0392326614432702696706022129, −16.03133927437081558364055614769, −15.19504515504866097791449294541, −14.48417568511339785590995811040, −13.82512893995958764062857662042, −12.67795399503489044400058127611, −12.03963345648620500337964855424, −11.609639062828895902602672027137, −10.09930236080750718355250378918, −9.421355192294187502084583966520, −8.20067789679514568244943213046, −7.831154827142969712887771565476, −7.00702362422582949051501317451, −6.029975954585758430786204374221, −4.785914039026064808992467544032, −3.64390490210019225498384408327, −3.07958046225628818930268968327, −1.75490175612564594561772707330, −0.65265324792234066561872910774, 1.344622833156255400035471179262, 2.77333362137831981367321241973, 3.58972732358377671464191164493, 4.30039084799584310449700635213, 5.15728446060195654090046346495, 6.50960064884461810946211430168, 7.50288416250981725514490211474, 8.23575506347299315837046131374, 9.0937233477040482139491244346, 9.829093574263040139227236919822, 10.77395950382042168308070341853, 11.76292986825951674711338870355, 12.18653384913451143140438074825, 13.68689662795963503901981803401, 14.27221070197411969040333958649, 14.96907796057517914761078139652, 15.79537672095260662228753043087, 16.48315885825530139225784796018, 17.1032652677495103560953523175, 18.55044480509920884413091069248, 19.19156585566728085638032643085, 19.86928394089211677174470367660, 20.42197201671998092482194803353, 21.459161244268540159694844252509, 22.140506663207756711081420559930

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