Properties

Label 1-896-896.411-r0-0-0
Degree $1$
Conductor $896$
Sign $0.891 - 0.453i$
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  + (0.442 + 0.896i)3-s + (−0.946 + 0.321i)5-s + (−0.608 + 0.793i)9-s + (−0.0654 − 0.997i)11-s + (−0.195 − 0.980i)13-s + (−0.707 − 0.707i)15-s + (−0.258 − 0.965i)17-s + (0.659 + 0.751i)19-s + (0.793 + 0.608i)23-s + (0.793 − 0.608i)25-s + (−0.980 − 0.195i)27-s + (−0.831 − 0.555i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s + (−0.321 − 0.946i)37-s + ⋯
L(s)  = 1  + (0.442 + 0.896i)3-s + (−0.946 + 0.321i)5-s + (−0.608 + 0.793i)9-s + (−0.0654 − 0.997i)11-s + (−0.195 − 0.980i)13-s + (−0.707 − 0.707i)15-s + (−0.258 − 0.965i)17-s + (0.659 + 0.751i)19-s + (0.793 + 0.608i)23-s + (0.793 − 0.608i)25-s + (−0.980 − 0.195i)27-s + (−0.831 − 0.555i)29-s + (−0.866 − 0.5i)31-s + (0.866 − 0.5i)33-s + (−0.321 − 0.946i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.891 - 0.453i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.891 - 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053317554 - 0.2523908331i\)
\(L(\frac12)\) \(\approx\) \(1.053317554 - 0.2523908331i\)
\(L(1)\) \(\approx\) \(0.9509480329 + 0.1239180553i\)
\(L(1)\) \(\approx\) \(0.9509480329 + 0.1239180553i\)

Euler product

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

−22.11524867520866097672808234281, −21.01063250797763910722704198864, −20.09174640315228378229083560597, −19.78869843708017916587521568207, −18.87388042074046135781270404797, −18.2664242379306899656957345707, −17.25667073164000255223199057681, −16.562436178885099830573830235866, −15.38799829424490213716176033844, −14.87391120553774129348236061204, −14.00539477554905240974789075475, −12.88928755128875558654553527754, −12.52115424699033827651864972441, −11.63107508438894722182513017098, −10.87509980590359701545912519720, −9.41290899002900865245077791804, −8.84644062341332904576009768560, −7.88776619363833976010136756129, −7.15084738186768874911053416800, −6.5754168383620311264715963676, −5.12997510630766002169844368456, −4.22899577360511428936040192124, −3.245962244823346744174483302940, −2.12065908238688473931282149029, −1.151497479783208252909984592494, 0.50597835029551658588304766740, 2.438957894903966080004159837657, 3.381016532796774440467009443637, 3.831032417743259338015557914750, 5.10628935727912140659669811409, 5.752212687199866365060069813436, 7.32827445231070525396726622932, 7.84148701271159165518565378232, 8.80822276333864478140428918107, 9.56327774050917705177259439423, 10.66009132432503248168172454479, 11.14595202441536514732383343555, 11.993121597841401911738337206797, 13.18942047693056330987449061808, 14.030155608436346149868790433647, 14.840313217372967282114903235221, 15.52840525378002123444429539255, 16.150616751026186922855063347861, 16.84109585186185814253610228675, 18.03881689732108240548243552939, 18.94157481061469773061723235959, 19.55859249278490514174064579370, 20.42604658385520152749210613943, 20.90904975910124713207874586477, 22.05577203593697907692485579898

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