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Properties

Label 1-896-896.573-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.740 - 0.671i$
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  + (−0.195 + 0.980i)3-s + (0.831 + 0.555i)5-s + (−0.923 − 0.382i)9-s + (−0.980 + 0.195i)11-s + (−0.831 + 0.555i)13-s + (−0.707 + 0.707i)15-s + (0.707 + 0.707i)17-s + (0.555 + 0.831i)19-s + (−0.382 + 0.923i)23-s + (0.382 + 0.923i)25-s + (0.555 − 0.831i)27-s + (0.980 + 0.195i)29-s − i·31-s − i·33-s + (0.555 − 0.831i)37-s + ⋯
L(s)  = 1  + (−0.195 + 0.980i)3-s + (0.831 + 0.555i)5-s + (−0.923 − 0.382i)9-s + (−0.980 + 0.195i)11-s + (−0.831 + 0.555i)13-s + (−0.707 + 0.707i)15-s + (0.707 + 0.707i)17-s + (0.555 + 0.831i)19-s + (−0.382 + 0.923i)23-s + (0.382 + 0.923i)25-s + (0.555 − 0.831i)27-s + (0.980 + 0.195i)29-s − i·31-s − i·33-s + (0.555 − 0.831i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.740 - 0.671i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (573, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.740 - 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4140138158 + 1.073290458i\)
\(L(\frac12)\) \(\approx\) \(-0.4140138158 + 1.073290458i\)
\(L(1)\) \(\approx\) \(0.7742914763 + 0.5787652266i\)
\(L(1)\) \(\approx\) \(0.7742914763 + 0.5787652266i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.195 + 0.980i)T \)
5 \( 1 + (0.831 + 0.555i)T \)
11 \( 1 + (-0.980 + 0.195i)T \)
13 \( 1 + (-0.831 + 0.555i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (0.555 + 0.831i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (0.980 + 0.195i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.555 - 0.831i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (0.195 + 0.980i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.980 - 0.195i)T \)
59 \( 1 + (-0.831 - 0.555i)T \)
61 \( 1 + (0.195 - 0.980i)T \)
67 \( 1 + (-0.195 + 0.980i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (-0.555 - 0.831i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
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

−21.17554284640546867984758162956, −20.41879813701527804562392513757, −19.75203729829547184266575455362, −18.64407601107062670345540673122, −18.15703684876816074378223272745, −17.37032756836810060365126584048, −16.715272223223037865686853419322, −15.822991905011292567014342981668, −14.664262669343229870055712958508, −13.712989597382312086152244313320, −13.31138709764312490126445662966, −12.39497948758686920525706993009, −11.83632816857360754952330112909, −10.59366284224179028415331552479, −9.87566738310529719942953950173, −8.82160223995959455973710595246, −7.94733082937578665942077474065, −7.22797840291533390324744114037, −6.15204033529544629193268038175, −5.38742240690779002088752178544, −4.74082344372525488928088323269, −2.815863044895487535728691112042, −2.398653856432317446199250647845, −1.02910206697361911119400958329, −0.26941652569777454001377463371, 1.56022945217336067894167899400, 2.72491196117947275344892933, 3.49551123072207750813977616656, 4.729275698996729033879726874605, 5.47063764388804844179223317253, 6.20734262010452951428341976136, 7.35911146431383875922314937994, 8.33793358208716998901654404598, 9.54374502167355097754530932559, 10.00765639751223373990631867689, 10.5989764964699569592164281618, 11.61020359234450277081572865893, 12.48469705437274574851046411804, 13.592313806042379781907077696030, 14.46530335347259017793314956167, 14.87537457205867750083619257144, 16.004719290682136998150602085798, 16.557914646768235775093470727828, 17.565958511907245656038332017052, 18.02681555789897235548893925337, 19.10079247264082817322393254957, 19.98542383419879092308229289452, 20.95488634860466532095835776279, 21.63274649457192429826407590032, 21.83210150292418207615561077123

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