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Properties

Label 1-1792-1792.397-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.643 - 0.765i$
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  + (0.973 + 0.227i)3-s + (−0.812 + 0.582i)5-s + (0.896 + 0.442i)9-s + (−0.0327 + 0.999i)11-s + (−0.995 − 0.0980i)13-s + (−0.923 + 0.382i)15-s + (−0.793 + 0.608i)17-s + (−0.935 − 0.352i)19-s + (0.946 − 0.321i)23-s + (0.321 − 0.946i)25-s + (0.773 + 0.634i)27-s + (0.471 − 0.881i)29-s + (−0.258 − 0.965i)31-s + (−0.258 + 0.965i)33-s + (−0.986 − 0.162i)37-s + ⋯
L(s)  = 1  + (0.973 + 0.227i)3-s + (−0.812 + 0.582i)5-s + (0.896 + 0.442i)9-s + (−0.0327 + 0.999i)11-s + (−0.995 − 0.0980i)13-s + (−0.923 + 0.382i)15-s + (−0.793 + 0.608i)17-s + (−0.935 − 0.352i)19-s + (0.946 − 0.321i)23-s + (0.321 − 0.946i)25-s + (0.773 + 0.634i)27-s + (0.471 − 0.881i)29-s + (−0.258 − 0.965i)31-s + (−0.258 + 0.965i)33-s + (−0.986 − 0.162i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.643 - 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.173755071 - 0.5463151573i\)
\(L(\frac12)\) \(\approx\) \(1.173755071 - 0.5463151573i\)
\(L(1)\) \(\approx\) \(1.069457327 + 0.2010299717i\)
\(L(1)\) \(\approx\) \(1.069457327 + 0.2010299717i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.973 + 0.227i)T \)
5 \( 1 + (-0.812 + 0.582i)T \)
11 \( 1 + (-0.0327 + 0.999i)T \)
13 \( 1 + (-0.995 - 0.0980i)T \)
17 \( 1 + (-0.793 + 0.608i)T \)
19 \( 1 + (-0.935 - 0.352i)T \)
23 \( 1 + (0.946 - 0.321i)T \)
29 \( 1 + (0.471 - 0.881i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (-0.986 - 0.162i)T \)
41 \( 1 + (0.980 + 0.195i)T \)
43 \( 1 + (-0.290 + 0.956i)T \)
47 \( 1 + (-0.991 + 0.130i)T \)
53 \( 1 + (-0.999 - 0.0327i)T \)
59 \( 1 + (0.412 - 0.910i)T \)
61 \( 1 + (-0.729 - 0.683i)T \)
67 \( 1 + (-0.227 + 0.973i)T \)
71 \( 1 + (0.831 + 0.555i)T \)
73 \( 1 + (-0.997 + 0.0654i)T \)
79 \( 1 + (0.608 - 0.793i)T \)
83 \( 1 + (-0.634 - 0.773i)T \)
89 \( 1 + (0.751 - 0.659i)T \)
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

−19.82633105344569134656056245875, −19.54763525968746015047272414911, −18.94564916289351461606349129845, −18.07431280595580284161991233286, −17.08713983573919501502687324786, −16.3160287310089509454567374370, −15.63941477505624962329826367723, −14.9468262155965259398274616585, −14.20463626948766831303280363714, −13.45726080792405255804742541444, −12.66157333654916971935576865398, −12.13047880267067002215406599760, −11.13954559932974618056318181758, −10.34831890951874188134145151656, −9.09684358190557182643940978631, −8.856656971300051182464693676508, −8.04008996750176227236578304852, −7.24165814981980201860608981697, −6.60310222489679105455253760619, −5.19007354271246470770387151366, −4.53545006318480781337164418030, −3.54996944582021581886666816414, −2.91730365321397694790795191211, −1.83673252647383824674390443813, −0.79408478768885715737250578771, 0.23948515494731448097685214161, 1.90228741150796726653920922823, 2.53136867877294326482912320578, 3.38031434257628497201130496625, 4.52001888010219866359526046970, 4.59747325748558099572210599118, 6.360132946666311732037117222189, 7.08727113066968041952989770862, 7.73874098336213640037737707485, 8.442699380793246301420893989108, 9.336883041339406000780206349599, 10.082474322270377199917512824719, 10.80002722906949228303509119899, 11.63630588942264864048063760167, 12.76003050991645339709599080527, 13.00361636944126177980974337332, 14.36296629200077596331079632855, 14.75118529604731431770573132415, 15.344822924664171744041842761650, 15.88320346457204977468571680893, 17.07105010854903231542655535441, 17.69786755580515894958619383407, 18.74391548711041912703938000573, 19.36817196360264008140675686935, 19.74962516876617779518947543626

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