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Properties

Label 1-1792-1792.1637-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  + (−0.881 + 0.471i)3-s + (−0.634 − 0.773i)5-s + (0.555 − 0.831i)9-s + (0.956 − 0.290i)11-s + (0.773 + 0.634i)13-s + (0.923 + 0.382i)15-s + (−0.923 + 0.382i)17-s + (−0.0980 − 0.995i)19-s + (0.980 − 0.195i)23-s + (−0.195 + 0.980i)25-s + (−0.0980 + 0.995i)27-s + (−0.290 + 0.956i)29-s + (−0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (−0.995 − 0.0980i)37-s + ⋯
L(s)  = 1  + (−0.881 + 0.471i)3-s + (−0.634 − 0.773i)5-s + (0.555 − 0.831i)9-s + (0.956 − 0.290i)11-s + (0.773 + 0.634i)13-s + (0.923 + 0.382i)15-s + (−0.923 + 0.382i)17-s + (−0.0980 − 0.995i)19-s + (0.980 − 0.195i)23-s + (−0.195 + 0.980i)25-s + (−0.0980 + 0.995i)27-s + (−0.290 + 0.956i)29-s + (−0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (−0.995 − 0.0980i)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: $\chi_{1792} (1637, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.844 + 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04994057386 + 0.1722120665i\)
\(L(\frac12)\) \(\approx\) \(0.04994057386 + 0.1722120665i\)
\(L(1)\) \(\approx\) \(0.6949072707 + 0.01208921951i\)
\(L(1)\) \(\approx\) \(0.6949072707 + 0.01208921951i\)

Euler product

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

−19.50711445846433406360512973880, −18.92224783527763223087351153010, −18.242738652550755920370914818472, −17.58061713360128244787618793562, −16.917240462013483915730398178737, −15.96756490013186999621229030441, −15.44147176990688242217884193629, −14.54599818907346799350002352210, −13.74122993327256055537735090060, −12.87773814891015797360643606147, −12.136135156076047543651430355189, −11.43270028467489194813441488887, −10.88904866753972657530133282588, −10.20157012602615233769010301275, −9.08733864736607473929014471279, −8.11093708598350363850029980266, −7.31910327281345398940732925307, −6.65355826796565455798526226625, −6.05237525839484723773222605727, −5.04311343564814429329891941273, −4.07310578885727391346318074616, −3.32533773760591821992936807584, −2.08939350949506877188669994554, −1.142212554097074503180486137281, −0.04864968835629453025329573480, 0.880265952757332231425103128638, 1.75182429298156880425808296227, 3.45626597257829847659729158868, 4.03422712642458737626257539812, 4.804474488047987726419970470913, 5.51671019969650737283343013479, 6.673820916847583409922366165244, 6.959864003698737576426747293485, 8.522184222496973777167911793955, 8.905119035634723936247714903744, 9.65757133121423508648248774681, 10.936412300772045858429739111653, 11.28112318781698022990313002003, 11.89588493248159202727665104963, 12.87606519133823634756197371932, 13.3574439188594518694169399280, 14.66058363803400253279321245750, 15.28654519041907102149870308736, 16.07635830477622951358260641593, 16.6157202400147707448348936566, 17.18610840203277723599028543529, 17.97410686120596393704170310328, 18.88368517179908138004605672884, 19.648249462471294186228670964681, 20.35125872375934423067031742936

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