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Properties

Label 1-1792-1792.1139-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.765 + 0.643i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.765 + 0.643i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.765 + 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5702616921 + 0.2080176385i\)
\(L(\frac12)\) \(\approx\) \(0.5702616921 + 0.2080176385i\)
\(L(1)\) \(\approx\) \(0.6119384868 + 0.01843272457i\)
\(L(1)\) \(\approx\) \(0.6119384868 + 0.01843272457i\)

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

−20.31642547630017155071351490631, −19.44015221392160361263767254421, −18.57640999835368136833705995247, −17.60400381416324306575062329413, −17.39592209535904202516242527210, −16.21389635478628918555545692936, −15.94884668990835571709757517733, −15.17458558389615325765636685338, −14.32407273731579111092607189899, −13.17108614708863694128478860997, −12.3978632636369601159106138948, −11.955394492019414042617522300939, −11.318594143008423216931020920422, −10.37008918205054187779177459484, −9.58709663731433294039995894358, −8.95449752086715942316425913771, −7.59432276641438797288159958307, −7.28240992781012714537838619261, −6.29132172946237538972776511157, −5.1921704954336926635176852005, −4.63756809979216743297198037943, −4.10441949565527297194722763217, −2.79547492884580247256647542791, −1.58610874160769602682539563360, −0.41467987078517399909416327456, 0.651485909429311750397909145849, 1.9496697305431449140325664991, 3.0610804589929745136671052496, 3.98568056102624536631031794905, 4.78880303628745341716798477021, 5.79399634318327285571864878244, 6.402492816994824727281108501811, 7.34885617923469037018241893560, 7.837148152676800207246861920482, 8.860790450149672610305339265473, 10.062760229836821210986499805677, 10.61999922884204456088021090812, 11.36732555954113248724599964349, 12.03125612282350117749002254790, 12.50701102644596522499449558812, 13.71634140740223336986528372625, 14.22223416967835339520740478115, 15.47209697810872265358541594388, 15.77168625418699777698362493737, 16.61189230561453214680899396266, 17.435959341037871675437826469829, 18.01152602056713553271892060972, 18.87758781359098351501972099738, 19.38599532762333860509016846111, 20.04284660464406545784717691685

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