Properties

Label 1-1792-1792.1581-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.844 + 0.534i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (−0.471 + 0.881i)3-s + (−0.773 − 0.634i)5-s + (−0.555 − 0.831i)9-s + (−0.290 + 0.956i)11-s + (−0.634 − 0.773i)13-s + (0.923 − 0.382i)15-s + (−0.923 − 0.382i)17-s + (0.995 + 0.0980i)19-s + (−0.980 − 0.195i)23-s + (0.195 + 0.980i)25-s + (0.995 − 0.0980i)27-s + (−0.956 + 0.290i)29-s + (−0.707 + 0.707i)31-s + (−0.707 − 0.707i)33-s + (−0.0980 − 0.995i)37-s + ⋯
L(s)  = 1  + (−0.471 + 0.881i)3-s + (−0.773 − 0.634i)5-s + (−0.555 − 0.831i)9-s + (−0.290 + 0.956i)11-s + (−0.634 − 0.773i)13-s + (0.923 − 0.382i)15-s + (−0.923 − 0.382i)17-s + (0.995 + 0.0980i)19-s + (−0.980 − 0.195i)23-s + (0.195 + 0.980i)25-s + (0.995 − 0.0980i)27-s + (−0.956 + 0.290i)29-s + (−0.707 + 0.707i)31-s + (−0.707 − 0.707i)33-s + (−0.0980 − 0.995i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.844 + 0.534i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1581, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.844 + 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3195929371 + 0.09268023432i\)
\(L(\frac12)\) \(\approx\) \(0.3195929371 + 0.09268023432i\)
\(L(1)\) \(\approx\) \(0.5520850599 + 0.08195467506i\)
\(L(1)\) \(\approx\) \(0.5520850599 + 0.08195467506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.471 + 0.881i)T \)
5 \( 1 + (-0.773 - 0.634i)T \)
11 \( 1 + (-0.290 + 0.956i)T \)
13 \( 1 + (-0.634 - 0.773i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
19 \( 1 + (0.995 + 0.0980i)T \)
23 \( 1 + (-0.980 - 0.195i)T \)
29 \( 1 + (-0.956 + 0.290i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (-0.0980 - 0.995i)T \)
41 \( 1 + (-0.195 + 0.980i)T \)
43 \( 1 + (-0.471 - 0.881i)T \)
47 \( 1 + (-0.382 + 0.923i)T \)
53 \( 1 + (-0.956 - 0.290i)T \)
59 \( 1 + (-0.634 + 0.773i)T \)
61 \( 1 + (-0.881 - 0.471i)T \)
67 \( 1 + (-0.881 - 0.471i)T \)
71 \( 1 + (0.555 - 0.831i)T \)
73 \( 1 + (-0.831 + 0.555i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (0.0980 - 0.995i)T \)
89 \( 1 + (0.980 - 0.195i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−19.765605460638792343354489651845, −19.08217175962701385508749226046, −18.55596866293910875706825608054, −17.95759932531569657481995130667, −17.0013621815676470368415755182, −16.355632192951933657176994334057, −15.60342924211564758327516844813, −14.70397754396692775818074190721, −13.87531715752023701870758076135, −13.35190413547168638409200102985, −12.35012919574873578667862198957, −11.572542447345146947970781672351, −11.282320658602876091296637979394, −10.40652252154877122299606229726, −9.31628061973299590850653579403, −8.26693279080746738229632473223, −7.69182132472417843619046753945, −6.95065291676416842528856222990, −6.25288953034547580138457660822, −5.42543301216505907224816870603, −4.37977225745507605450981661050, −3.408433145000414984430992580139, −2.4816317553964893233667874063, −1.58317486923952784414941866069, −0.22297151206543569248698819544, 0.25809490062502723964754078193, 1.662882800888695931687712492070, 2.97911018146014826686434420695, 3.79877428946750446737403305535, 4.74651098283691854489850953447, 5.05931366741021077629466558780, 6.05749577683948265151707791283, 7.27498171868027954747863401823, 7.806609505285345909236708309785, 8.94919635416974242845102179441, 9.48280858648975610480045803808, 10.33138778174960073762043867699, 11.06729273521844203429024351903, 11.9228558365559869784669540597, 12.41074109976146592427413350871, 13.22170144839832176393732840112, 14.456919816816642075940916991829, 15.07157652770106320193770135449, 15.83843385313467928954359262893, 16.1639136635001669848548999075, 17.10359047118937491893119863373, 17.79613036999005224375519233147, 18.38831586150820091216829822538, 19.81953757459966055780708873991, 20.1230977896176365326702349617

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