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Properties

Label 1-896-896.227-r0-0-0
Degree $1$
Conductor $896$
Sign $0.759 - 0.650i$
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  + (0.321 + 0.946i)3-s + (−0.442 − 0.896i)5-s + (−0.793 + 0.608i)9-s + (0.751 − 0.659i)11-s + (0.555 + 0.831i)13-s + (0.707 − 0.707i)15-s + (0.258 − 0.965i)17-s + (−0.0654 − 0.997i)19-s + (−0.608 − 0.793i)23-s + (−0.608 + 0.793i)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 + (−0.896 + 0.442i)37-s + ⋯
L(s)  = 1  + (0.321 + 0.946i)3-s + (−0.442 − 0.896i)5-s + (−0.793 + 0.608i)9-s + (0.751 − 0.659i)11-s + (0.555 + 0.831i)13-s + (0.707 − 0.707i)15-s + (0.258 − 0.965i)17-s + (−0.0654 − 0.997i)19-s + (−0.608 − 0.793i)23-s + (−0.608 + 0.793i)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 + (−0.896 + 0.442i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.759 - 0.650i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.759 - 0.650i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.260224709 - 0.4655492180i\)
\(L(\frac12)\) \(\approx\) \(1.260224709 - 0.4655492180i\)
\(L(1)\) \(\approx\) \(1.080284709 + 0.006943531745i\)
\(L(1)\) \(\approx\) \(1.080284709 + 0.006943531745i\)

Euler product

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

−22.29858066997304034981323009048, −21.20257408553360636701655033479, −20.14005796458870988158233294873, −19.64506650451148048863414118595, −18.9183242104387783087810884954, −18.053195786416589645927169554029, −17.65753560433545317118025306619, −16.53969243550788549793990241510, −15.41857531136324480482968786537, −14.64425430240249948717023046611, −14.237510458058946337960974169102, −13.08761693148294609832475714510, −12.40043061846020383442344737474, −11.643369200671573544518300262591, −10.73634288001771644759735717535, −9.85120642962484416984744024663, −8.68167685913186573819206830600, −7.86470286860963416394194429521, −7.24817784450471900435900172958, −6.32364015923454475450540387114, −5.64961606573194867084697153250, −3.88282761826387080478326649177, −3.40805210422989458244039266411, −2.17088142932311873596716483330, −1.27775289991316849658555472553, 0.61345168681470523377491300513, 2.09679954706655974306790313613, 3.38090285554937871886231967214, 4.129469710518078408988512797828, 4.85546485271987780859692357527, 5.798697835245538018536799273568, 6.96078548761686643999067617292, 8.1622662039249505802452957163, 8.92505877008451052834068220744, 9.28614839759062243353662182235, 10.431281467973972847564826221746, 11.48274505817723784034697307288, 11.83272580111408700699768068470, 13.16239255180814071006827177543, 13.95592574944867303025341779883, 14.6093260239788102559502891347, 15.863412074607014849542348942532, 16.07345920660293519648195569816, 16.83953894980386040452004370443, 17.704292286011682985351718610706, 19.08007337524867442764734918331, 19.4955596119292907924521302874, 20.54934550221375120025308543438, 20.85500710974635530386388134364, 21.821476613721742070147365889

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