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Properties

Label 1-1792-1792.851-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.877 - 0.480i$
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  + (−0.999 + 0.0327i)3-s + (0.935 − 0.352i)5-s + (0.997 − 0.0654i)9-s + (0.973 − 0.227i)11-s + (−0.634 − 0.773i)13-s + (−0.923 + 0.382i)15-s + (−0.130 − 0.991i)17-s + (0.582 − 0.812i)19-s + (−0.659 + 0.751i)23-s + (0.751 − 0.659i)25-s + (−0.995 + 0.0980i)27-s + (0.956 − 0.290i)29-s + (0.965 + 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.910 − 0.412i)37-s + ⋯
L(s)  = 1  + (−0.999 + 0.0327i)3-s + (0.935 − 0.352i)5-s + (0.997 − 0.0654i)9-s + (0.973 − 0.227i)11-s + (−0.634 − 0.773i)13-s + (−0.923 + 0.382i)15-s + (−0.130 − 0.991i)17-s + (0.582 − 0.812i)19-s + (−0.659 + 0.751i)23-s + (0.751 − 0.659i)25-s + (−0.995 + 0.0980i)27-s + (0.956 − 0.290i)29-s + (0.965 + 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.910 − 0.412i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.877 - 0.480i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.877 - 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3434514558 - 1.341902355i\)
\(L(\frac12)\) \(\approx\) \(0.3434514558 - 1.341902355i\)
\(L(1)\) \(\approx\) \(0.8981138382 - 0.2808290560i\)
\(L(1)\) \(\approx\) \(0.8981138382 - 0.2808290560i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.999 + 0.0327i)T \)
5 \( 1 + (0.935 - 0.352i)T \)
11 \( 1 + (0.973 - 0.227i)T \)
13 \( 1 + (-0.634 - 0.773i)T \)
17 \( 1 + (-0.130 - 0.991i)T \)
19 \( 1 + (0.582 - 0.812i)T \)
23 \( 1 + (-0.659 + 0.751i)T \)
29 \( 1 + (0.956 - 0.290i)T \)
31 \( 1 + (0.965 + 0.258i)T \)
37 \( 1 + (-0.910 - 0.412i)T \)
41 \( 1 + (0.195 - 0.980i)T \)
43 \( 1 + (-0.471 - 0.881i)T \)
47 \( 1 + (-0.608 - 0.793i)T \)
53 \( 1 + (-0.227 - 0.973i)T \)
59 \( 1 + (-0.986 - 0.162i)T \)
61 \( 1 + (0.849 - 0.528i)T \)
67 \( 1 + (0.0327 + 0.999i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (-0.896 - 0.442i)T \)
79 \( 1 + (-0.991 - 0.130i)T \)
83 \( 1 + (-0.0980 + 0.995i)T \)
89 \( 1 + (0.321 - 0.946i)T \)
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

−20.41983529553444070067200732213, −19.40340138770502769511331322041, −18.77599230891372404780358514174, −17.921863956316460680562878453015, −17.41397870567736761864501157428, −16.771079405838977930157801081449, −16.20582281549790546303040747460, −15.06860964262627039344810399282, −14.35788598321283064057138237682, −13.73393874167286106182485820873, −12.68610064927403892481334799796, −12.10100822444501737844408914923, −11.43101774122960430383328412108, −10.41322347874312664138498248590, −9.97473160970111425314839194893, −9.24462889082460041981965252031, −8.11735291204865364416760927652, −7.023103605760949938744479465811, −6.33412344232601933017145659033, −5.99598222898851365510485373112, −4.79635694865147353769399904651, −4.24755616027800859137485273582, −2.98203322179089000558974006770, −1.74261506520909491372715328132, −1.28218274388031677615911636962, 0.308522731086935205675692182386, 1.05260121346584725009519109775, 2.05795930532012508815360383197, 3.169058924004598524797602423405, 4.372794072353830704425082184315, 5.190403164373313601734207457386, 5.64133449618321825875735631538, 6.66545556349475412433023618871, 7.14022996202199151984142209919, 8.3979356923736974662641953085, 9.362599783622703727288763271105, 9.91852831111533887162257556512, 10.59106017000103698424815176685, 11.76394233872698071790121185039, 11.96071099219327679950760182790, 13.014357735984863027278745047246, 13.700081731830571441619362231781, 14.360787158230727497784222303197, 15.59998643131183573711737345837, 16.02309155533049400800196456361, 16.996500082313058319991270851143, 17.6297949331433382940151898203, 17.76973021335239529791922743263, 18.887647092992454492841854717214, 19.75405330374529402026147020198

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