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Properties

Label 1-1792-1792.691-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  + (−0.227 + 0.973i)3-s + (−0.582 − 0.812i)5-s + (−0.896 − 0.442i)9-s + (0.999 + 0.0327i)11-s + (−0.0980 + 0.995i)13-s + (0.923 − 0.382i)15-s + (−0.793 + 0.608i)17-s + (0.352 − 0.935i)19-s + (0.946 − 0.321i)23-s + (−0.321 + 0.946i)25-s + (0.634 − 0.773i)27-s + (−0.881 − 0.471i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.162 − 0.986i)37-s + ⋯
L(s)  = 1  + (−0.227 + 0.973i)3-s + (−0.582 − 0.812i)5-s + (−0.896 − 0.442i)9-s + (0.999 + 0.0327i)11-s + (−0.0980 + 0.995i)13-s + (0.923 − 0.382i)15-s + (−0.793 + 0.608i)17-s + (0.352 − 0.935i)19-s + (0.946 − 0.321i)23-s + (−0.321 + 0.946i)25-s + (0.634 − 0.773i)27-s + (−0.881 − 0.471i)29-s + (0.258 + 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.162 − 0.986i)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: $\chi_{1792} (691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.765 + 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.119181759 + 0.4082503699i\)
\(L(\frac12)\) \(\approx\) \(1.119181759 + 0.4082503699i\)
\(L(1)\) \(\approx\) \(0.8887357176 + 0.1817812173i\)
\(L(1)\) \(\approx\) \(0.8887357176 + 0.1817812173i\)

Euler product

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

−20.06825256248629368880150008255, −19.249907668433982919750211154904, −18.61469682535599520091157090618, −18.13612328044183457368844770839, −17.18967428591628662353832475826, −16.71803385644855859064368990558, −15.47368573849027534005895997302, −14.98981361537645085229176573028, −14.07747049563348519773179839844, −13.513063838780948309163637071684, −12.5376358374765713867210476049, −11.91194382008645049018110886098, −11.228457447066724758455350083, −10.63079331858309463786183454612, −9.49946294532216350991115689472, −8.59366668067703290062838139610, −7.648418013623876527448731629474, −7.252061955000882316357452929729, −6.37158947036198816213734550985, −5.72966041959788040874179699066, −4.59506171784275120470522095660, −3.46053607305445943313576235536, −2.82068505693717475581480723519, −1.743434301799742675888585320871, −0.66721001579889512556080917159, 0.74797106342274435918011395503, 1.978525476737463684689119202763, 3.29207420731689050962932050679, 4.14237711410661958037912102484, 4.56626775680982851686100064362, 5.422807494150467425053242373292, 6.45686324749765162260894406789, 7.22237180755417854914416597767, 8.47558888552898705493087677727, 9.122557198336764789084590939532, 9.37773345282364128369475103787, 10.687390698284993300017161628562, 11.30857625785075373206610574440, 11.901487855439239091575028138510, 12.69898715397798665192128669186, 13.692033409532883234542190365221, 14.51629053651851149976355836475, 15.30619508006216007833417680964, 15.81409088209479396887646011806, 16.75228443995387171942936258089, 17.003708215999614487661225212551, 17.83153115665950837208061720371, 19.11418225799507075374860781349, 19.682386513350140038980297502873, 20.2451812189902315934345915490

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