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Properties

Label 1-1792-1792.1389-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.937 + 0.348i$
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  + (0.0327 + 0.999i)3-s + (−0.352 − 0.935i)5-s + (−0.997 + 0.0654i)9-s + (−0.227 − 0.973i)11-s + (0.773 − 0.634i)13-s + (0.923 − 0.382i)15-s + (0.130 + 0.991i)17-s + (−0.812 − 0.582i)19-s + (−0.659 + 0.751i)23-s + (−0.751 + 0.659i)25-s + (−0.0980 − 0.995i)27-s + (−0.290 − 0.956i)29-s + (0.965 + 0.258i)31-s + (0.965 − 0.258i)33-s + (0.412 − 0.910i)37-s + ⋯
L(s)  = 1  + (0.0327 + 0.999i)3-s + (−0.352 − 0.935i)5-s + (−0.997 + 0.0654i)9-s + (−0.227 − 0.973i)11-s + (0.773 − 0.634i)13-s + (0.923 − 0.382i)15-s + (0.130 + 0.991i)17-s + (−0.812 − 0.582i)19-s + (−0.659 + 0.751i)23-s + (−0.751 + 0.659i)25-s + (−0.0980 − 0.995i)27-s + (−0.290 − 0.956i)29-s + (0.965 + 0.258i)31-s + (0.965 − 0.258i)33-s + (0.412 − 0.910i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.937 + 0.348i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.937 + 0.348i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008119525957 + 0.04514001868i\)
\(L(\frac12)\) \(\approx\) \(0.008119525957 + 0.04514001868i\)
\(L(1)\) \(\approx\) \(0.8411480285 + 0.02226063070i\)
\(L(1)\) \(\approx\) \(0.8411480285 + 0.02226063070i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.0327 + 0.999i)T \)
5 \( 1 + (-0.352 - 0.935i)T \)
11 \( 1 + (-0.227 - 0.973i)T \)
13 \( 1 + (0.773 - 0.634i)T \)
17 \( 1 + (0.130 + 0.991i)T \)
19 \( 1 + (-0.812 - 0.582i)T \)
23 \( 1 + (-0.659 + 0.751i)T \)
29 \( 1 + (-0.290 - 0.956i)T \)
31 \( 1 + (0.965 + 0.258i)T \)
37 \( 1 + (0.412 - 0.910i)T \)
41 \( 1 + (0.195 - 0.980i)T \)
43 \( 1 + (-0.881 + 0.471i)T \)
47 \( 1 + (-0.608 - 0.793i)T \)
53 \( 1 + (0.973 - 0.227i)T \)
59 \( 1 + (0.162 - 0.986i)T \)
61 \( 1 + (0.528 + 0.849i)T \)
67 \( 1 + (-0.999 + 0.0327i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (-0.896 - 0.442i)T \)
79 \( 1 + (0.991 + 0.130i)T \)
83 \( 1 + (0.995 + 0.0980i)T \)
89 \( 1 + (0.321 - 0.946i)T \)
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

−19.562038039363308401466007676923, −18.77346893359210788586328477637, −18.31492113365645278036590642367, −17.849580356352626738421276959596, −16.781736848777252743995943902967, −16.06854181839956593955764343670, −14.98480541477161569372369986176, −14.54757972175781793177699113836, −13.72222887082726898135270889654, −13.06412984474672906480334424188, −12.03582401759880383085690931295, −11.707147718563598267428298623942, −10.739765361796297622173709826896, −10.01335271380781714529751226020, −8.938717096288898568950820084919, −8.04368112368173217889156181848, −7.46868910517043671633319570189, −6.52785970853213493485759666329, −6.30710242078345545065023165751, −4.950917521953096019953599881389, −3.98328105410442625538529515867, −2.944835010457507857184408024883, −2.23809343838117306725085774993, −1.32608400763082673480761748028, −0.01002319759657191801233795844, 0.83497581511570531686099263254, 2.17123647562152666974370081305, 3.430377737831501022811382228638, 3.90490192929316682458117213, 4.78161919160627930015670913816, 5.698931918557127640873022328569, 6.129214941792198605746341641792, 7.72248424473946774441296482511, 8.512380206278491335892094776282, 8.73903498706678003070991829792, 9.85797523500244879289100908492, 10.54950644057338156958007207935, 11.32113675971227803353196919649, 11.954164667771150804617191340172, 13.13254585442691074086656929559, 13.475133862825408352411118794518, 14.60824646593949336786307353796, 15.424807576648813052248437603335, 15.859117825731752116270745099250, 16.5689854229894620538688652366, 17.22835515477570603190064008371, 17.93028077440723605386289690288, 19.290286725420660374914183224649, 19.555729516349151523180580849228, 20.47584502870424435643901027597

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