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Properties

Label 1-1792-1792.1283-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.999 + 0.0388i$
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  + (0.986 − 0.162i)3-s + (0.227 + 0.973i)5-s + (0.946 − 0.321i)9-s + (−0.412 + 0.910i)11-s + (−0.290 − 0.956i)13-s + (0.382 + 0.923i)15-s + (−0.608 − 0.793i)17-s + (−0.0327 − 0.999i)19-s + (0.442 − 0.896i)23-s + (−0.896 + 0.442i)25-s + (0.881 − 0.471i)27-s + (−0.0980 + 0.995i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (−0.528 + 0.849i)37-s + ⋯
L(s)  = 1  + (0.986 − 0.162i)3-s + (0.227 + 0.973i)5-s + (0.946 − 0.321i)9-s + (−0.412 + 0.910i)11-s + (−0.290 − 0.956i)13-s + (0.382 + 0.923i)15-s + (−0.608 − 0.793i)17-s + (−0.0327 − 0.999i)19-s + (0.442 − 0.896i)23-s + (−0.896 + 0.442i)25-s + (0.881 − 0.471i)27-s + (−0.0980 + 0.995i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (−0.528 + 0.849i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.999 + 0.0388i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.999 + 0.0388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.458049891 + 0.06714908486i\)
\(L(\frac12)\) \(\approx\) \(3.458049891 + 0.06714908486i\)
\(L(1)\) \(\approx\) \(1.571662864 + 0.08349238008i\)
\(L(1)\) \(\approx\) \(1.571662864 + 0.08349238008i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.986 - 0.162i)T \)
5 \( 1 + (0.227 + 0.973i)T \)
11 \( 1 + (-0.412 + 0.910i)T \)
13 \( 1 + (-0.290 - 0.956i)T \)
17 \( 1 + (-0.608 - 0.793i)T \)
19 \( 1 + (-0.0327 - 0.999i)T \)
23 \( 1 + (0.442 - 0.896i)T \)
29 \( 1 + (-0.0980 + 0.995i)T \)
31 \( 1 + (0.258 + 0.965i)T \)
37 \( 1 + (-0.528 + 0.849i)T \)
41 \( 1 + (0.831 - 0.555i)T \)
43 \( 1 + (0.634 - 0.773i)T \)
47 \( 1 + (0.130 + 0.991i)T \)
53 \( 1 + (0.910 + 0.412i)T \)
59 \( 1 + (0.683 + 0.729i)T \)
61 \( 1 + (0.935 + 0.352i)T \)
67 \( 1 + (-0.162 - 0.986i)T \)
71 \( 1 + (-0.195 - 0.980i)T \)
73 \( 1 + (0.659 - 0.751i)T \)
79 \( 1 + (-0.793 - 0.608i)T \)
83 \( 1 + (0.471 - 0.881i)T \)
89 \( 1 + (0.997 + 0.0654i)T \)
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

−20.034272398666381177291017532602, −19.19440148217754421181229119588, −18.97070771724419272251259284985, −17.76263150733812791329982763803, −16.94773898865425434373203332766, −16.245788343363265853315272597350, −15.67970626077492909552434314158, −14.75277801500593686469892160073, −14.03844035460217152057037053254, −13.28662590396147452238015742222, −12.891969712848898220430273573574, −11.83996052332487735855056291158, −11.00321879966417904954739258240, −9.890787724727036043099583415988, −9.43952845136710655684276462802, −8.51593265399684598810435510165, −8.16266024681995367094987335677, −7.19467446649826983137664814461, −6.05973577760614507954853798433, −5.301039072552341478627112147698, −4.16287099730121990234817910692, −3.816208906621173396491266398455, −2.47091435825925396042691509574, −1.82209851741269487038501825012, −0.77276972306130976020720207422, 0.67614814422957167879257444582, 2.03735241036960384193904255287, 2.692674865353080917882422669190, 3.22029506674042989066459714666, 4.44107313022222574727856986095, 5.19295623430925323210069835846, 6.52331436127139306657577593434, 7.16201615034696995792720220026, 7.61911197484880994748864687002, 8.76344791029755158535879615670, 9.338404143695459806751135006518, 10.43226002906768204222423813137, 10.59483361858781067262211624143, 11.93167521033433073291354797968, 12.758302593209621729692790592569, 13.40377656246446615757046780927, 14.167735718892433611083226384029, 14.85248143551815003041867668463, 15.40134605431608836541503551859, 16.008103523654643999544716895, 17.4728693561642607521411145716, 17.90199718456055710137577326784, 18.530220540148054954117667991438, 19.355813024701040346224994251239, 20.02736787816219247028667913621

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