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Properties

Label 1-896-896.171-r0-0-0
Degree $1$
Conductor $896$
Sign $0.572 + 0.819i$
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  + (−0.751 + 0.659i)3-s + (−0.0654 − 0.997i)5-s + (0.130 − 0.991i)9-s + (0.321 + 0.946i)11-s + (0.831 + 0.555i)13-s + (0.707 + 0.707i)15-s + (−0.965 − 0.258i)17-s + (0.442 + 0.896i)19-s + (−0.991 − 0.130i)23-s + (−0.991 + 0.130i)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 + (0.997 − 0.0654i)37-s + ⋯
L(s)  = 1  + (−0.751 + 0.659i)3-s + (−0.0654 − 0.997i)5-s + (0.130 − 0.991i)9-s + (0.321 + 0.946i)11-s + (0.831 + 0.555i)13-s + (0.707 + 0.707i)15-s + (−0.965 − 0.258i)17-s + (0.442 + 0.896i)19-s + (−0.991 − 0.130i)23-s + (−0.991 + 0.130i)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 + (0.997 − 0.0654i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.572 + 0.819i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.572 + 0.819i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8848757595 + 0.4613711822i\)
\(L(\frac12)\) \(\approx\) \(0.8848757595 + 0.4613711822i\)
\(L(1)\) \(\approx\) \(0.8309916949 + 0.1380500950i\)
\(L(1)\) \(\approx\) \(0.8309916949 + 0.1380500950i\)

Euler product

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

−22.090216221755631944134126976327, −21.33530153773567409873691192324, −19.92686501420987449879347747157, −19.42417360021707828096249726095, −18.39602504289260878102283610393, −18.07102641493802743623298901057, −17.268890161559743022154199248430, −16.23005241405622066853594597617, −15.58525095236884496474893718336, −14.56333520466989554633412271537, −13.47941001050862894706985017492, −13.27327352739229691601342540936, −11.86716143837480957696900681432, −11.249434701005701262591858560554, −10.78748311497862978266961808925, −9.74118671997777448518559104420, −8.432635400953551211821573112269, −7.7322947199132331775583923758, −6.5881755932403276571993626689, −6.27082615079411439026885748210, −5.28675946264742888385137187978, −3.99407770383174277503350361540, −2.97869561873117260591459106732, −1.93046548002453477722105178050, −0.60982070796597518333135799537, 0.999501127584761548850272316, 2.10795535014035487612380800057, 4.02797903364770454244161650440, 4.14493755848135513222952027492, 5.31438370345092735636646487195, 6.046205523961398752847171256198, 7.03786453127572304311913051646, 8.223196368776784330044132371233, 9.19511041840866695912793808183, 9.7011605388344797814608811581, 10.71470857673490800622269645014, 11.70108580704314665917517823811, 12.15834424767065461731034532668, 13.09368321804675902839078716088, 14.04052693610405461144818141058, 15.15080584053972055949992470153, 15.86339614423262029774151723849, 16.45680179315961051156532362448, 17.22218310097678503666185401446, 17.90504091197653397025512505879, 18.77839994198757553716576114873, 20.10601941695336077339349250823, 20.516419331869494422614628696741, 21.16618829059744374936309742161, 22.234294360135985974181250178445

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