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Properties

Label 1-896-896.475-r0-0-0
Degree $1$
Conductor $896$
Sign $-0.941 + 0.336i$
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  + (−0.555 + 0.831i)3-s + (−0.195 + 0.980i)5-s + (−0.382 − 0.923i)9-s + (0.831 − 0.555i)11-s + (0.195 + 0.980i)13-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)17-s + (0.980 − 0.195i)19-s + (−0.923 + 0.382i)23-s + (−0.923 − 0.382i)25-s + (0.980 + 0.195i)27-s + (0.831 + 0.555i)29-s − i·31-s − i·33-s + (−0.980 − 0.195i)37-s + ⋯
L(s)  = 1  + (−0.555 + 0.831i)3-s + (−0.195 + 0.980i)5-s + (−0.382 − 0.923i)9-s + (0.831 − 0.555i)11-s + (0.195 + 0.980i)13-s + (−0.707 − 0.707i)15-s + (−0.707 + 0.707i)17-s + (0.980 − 0.195i)19-s + (−0.923 + 0.382i)23-s + (−0.923 − 0.382i)25-s + (0.980 + 0.195i)27-s + (0.831 + 0.555i)29-s − i·31-s − i·33-s + (−0.980 − 0.195i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.941 + 0.336i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ -0.941 + 0.336i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1536213382 + 0.8853415875i\)
\(L(\frac12)\) \(\approx\) \(0.1536213382 + 0.8853415875i\)
\(L(1)\) \(\approx\) \(0.7000640761 + 0.4514952680i\)
\(L(1)\) \(\approx\) \(0.7000640761 + 0.4514952680i\)

Euler product

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

−21.76828025373301486706763131794, −20.397461524036510087539780695740, −20.17034619557641159114885005520, −19.29782957685637582378381248594, −18.24516239167435794370130131725, −17.63650121860329323113075294416, −16.98895023812235160592037369380, −16.09823731651270083869914538431, −15.43854218250807332611537927553, −14.09246609147820324274365117846, −13.46906167639266680568575282, −12.569268970139720052123478738381, −12.011220994936068531146363655957, −11.35862878851084115328031466110, −10.1702790283197840690735282650, −9.246626909910334719209412879040, −8.24653437866052086393751095613, −7.59376043167308704041570966897, −6.606318266919462929626052928148, −5.69578387420850443354191421417, −4.90569009946958230477783463632, −3.95222014975052956740323630676, −2.49436452873739189271055064045, −1.42365508267684058463396045863, −0.46392417072854087603839094890, 1.42791998270297626802829171893, 2.91340467423725093158244269249, 3.77212365329594969875754768117, 4.447931408502940149428990077608, 5.74540984989919955226759951404, 6.434654990236840844002018971103, 7.17125503214282813733263806102, 8.55411210089229690024353944432, 9.30233279365219326080222984530, 10.225955016692453186990110349839, 10.98721940144561004149061284844, 11.589685641821451137659717558549, 12.28565545059790399322264801043, 13.880401671881188352909740405748, 14.25417505957762570288879632281, 15.207531081031475095321316823438, 16.013877240035727263981414084784, 16.529136603448361789150236874232, 17.79627629295070581103603898991, 17.96803084698265937797323584821, 19.42605760921933590221435282426, 19.655232994316952094879994708754, 21.04816189857632482740650442493, 21.63874741010203441924167720518, 22.279084964155329880058264574677

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