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Properties

Label 1-1792-1792.563-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.416 - 0.909i$
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  + (0.729 + 0.683i)3-s + (0.412 − 0.910i)5-s + (0.0654 + 0.997i)9-s + (0.528 − 0.849i)11-s + (0.0980 − 0.995i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (0.986 − 0.162i)19-s + (−0.751 − 0.659i)23-s + (−0.659 − 0.751i)25-s + (−0.634 + 0.773i)27-s + (0.881 + 0.471i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (0.935 − 0.352i)37-s + ⋯
L(s)  = 1  + (0.729 + 0.683i)3-s + (0.412 − 0.910i)5-s + (0.0654 + 0.997i)9-s + (0.528 − 0.849i)11-s + (0.0980 − 0.995i)13-s + (0.923 − 0.382i)15-s + (−0.130 − 0.991i)17-s + (0.986 − 0.162i)19-s + (−0.751 − 0.659i)23-s + (−0.659 − 0.751i)25-s + (−0.634 + 0.773i)27-s + (0.881 + 0.471i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (0.935 − 0.352i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.416 - 0.909i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.416 - 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.795009021 - 1.152242302i\)
\(L(\frac12)\) \(\approx\) \(1.795009021 - 1.152242302i\)
\(L(1)\) \(\approx\) \(1.420562292 - 0.2098991549i\)
\(L(1)\) \(\approx\) \(1.420562292 - 0.2098991549i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.729 + 0.683i)T \)
5 \( 1 + (0.412 - 0.910i)T \)
11 \( 1 + (0.528 - 0.849i)T \)
13 \( 1 + (0.0980 - 0.995i)T \)
17 \( 1 + (-0.130 - 0.991i)T \)
19 \( 1 + (0.986 - 0.162i)T \)
23 \( 1 + (-0.751 - 0.659i)T \)
29 \( 1 + (0.881 + 0.471i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (0.935 - 0.352i)T \)
41 \( 1 + (-0.980 - 0.195i)T \)
43 \( 1 + (-0.956 - 0.290i)T \)
47 \( 1 + (-0.608 - 0.793i)T \)
53 \( 1 + (-0.849 - 0.528i)T \)
59 \( 1 + (-0.812 + 0.582i)T \)
61 \( 1 + (-0.973 - 0.227i)T \)
67 \( 1 + (-0.683 + 0.729i)T \)
71 \( 1 + (0.831 + 0.555i)T \)
73 \( 1 + (-0.442 + 0.896i)T \)
79 \( 1 + (0.991 + 0.130i)T \)
83 \( 1 + (-0.773 + 0.634i)T \)
89 \( 1 + (0.946 + 0.321i)T \)
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

−20.00759598682476986250840646824, −19.72574188007927035708962013614, −18.78434194970420556168411383012, −18.23781841648373920643752438022, −17.62777321685061308902608049873, −16.84697809085267191805683168513, −15.679857907481170613904508906545, −14.94379962832752520194780763290, −14.345465876105037423674934338, −13.79387746909544381102936798702, −13.06544914781532306354508332226, −12.06100976118025844558044183061, −11.582216342804640305619716992981, −10.447659144929543530372051374939, −9.5989366580099388451082690663, −9.17071741580809876181214166233, −7.94802832021941118402919279942, −7.44324239878946133987239449737, −6.409509628502998602166034553926, −6.24548072596422338612463576932, −4.71688571274617682525619070265, −3.6951773833131594914901605859, −3.04721275340196258343108925776, −1.81916967510651306932433833569, −1.62199168299903526622080299141, 0.6492424540481892564301983815, 1.77900161277007099398285581633, 2.88219622866137529836557659639, 3.54145818224733806531798988017, 4.59568288856748237920641549855, 5.24075005777071581646119602737, 5.99892396433484180082691714590, 7.23939248681594047110865109191, 8.24110699607698361988498057215, 8.65167205836017658291762250057, 9.51867791114163259815013254685, 10.0218114962650692807758575797, 10.98956638940901809377249504489, 11.838287385421973295819009090763, 12.76632720040422952973513697814, 13.62371279086678477751212580232, 13.97583874935345393249300625902, 14.88293281658920321689400599597, 15.812428093577268496544349011381, 16.31111102564531397990702022956, 16.85297270001889380051276132543, 17.95450326176238795667274327850, 18.52863042722948413007132187242, 19.77566169852407929618033486594, 20.12107487658692733107957224292

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