Properties

Label 1-896-896.515-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: $\chi_{896} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.148 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7483640939 + 0.8693837301i\)
\(L(\frac12)\) \(\approx\) \(0.7483640939 + 0.8693837301i\)
\(L(1)\) \(\approx\) \(0.7028797049 + 0.2332458127i\)
\(L(1)\) \(\approx\) \(0.7028797049 + 0.2332458127i\)

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

−21.79195256841591694647464938085, −20.621484186098410928301612382971, −19.941652187438542557819130139304, −19.02981140660986729939530572389, −18.48712661052283669178951977355, −17.412046876691648913340467461644, −16.77724362308600725739532148916, −16.09536336377554145936245325315, −15.38809421881028034221884359193, −14.28730767563912256579847153467, −13.30449542781668834052708426695, −12.462639732596422324567170414757, −11.75397368628612877756883082618, −11.28668071698071092125706021743, −10.25613031078639272762876878297, −9.22644781104303856652281732255, −8.229982152144445024624266332396, −7.32309694966935446985740200524, −6.69845119651422581491554413541, −5.42850410294447792237748634927, −4.89926798661053818224309460807, −3.836260256577965923005087192770, −2.696137533576152171473029347681, −1.07283142256145918086383635136, −0.48486215661724895939478797458, 0.77056102459072608941289189815, 2.08219691694861395131345551627, 3.588789421754587266656016275053, 4.21903017561992545798821771664, 5.110561625020084892342467026296, 6.1842557472218070478048835583, 7.11946960345014693369867319555, 7.58159661279627849832845453723, 9.07448126066696213818998826385, 9.78843348721731965967211709502, 10.7433826806532418441840335928, 11.50538893678572273886877637152, 12.067422661104508384987478550088, 12.797954709392327859516421572247, 14.12675851812982650768647762704, 15.07086318084814562304378718677, 15.4761354121717795171124189116, 16.46955034548171046095958085976, 17.226739945551181279032019495062, 17.81536202956265584773209063277, 18.903700249099905595895946420963, 19.47153759795786637178582477381, 20.375516947755532057286884858890, 21.37015950587974838499622533723, 22.22305034196412698096043850363

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