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Properties

Label 1-1792-1792.733-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.780 - 0.624i$
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  + (0.582 − 0.812i)3-s + (−0.528 + 0.849i)5-s + (−0.321 − 0.946i)9-s + (0.935 − 0.352i)11-s + (−0.881 − 0.471i)13-s + (0.382 + 0.923i)15-s + (−0.608 − 0.793i)17-s + (0.729 + 0.683i)19-s + (0.896 + 0.442i)23-s + (−0.442 − 0.896i)25-s + (−0.956 − 0.290i)27-s + (0.634 + 0.773i)29-s + (0.258 + 0.965i)31-s + (0.258 − 0.965i)33-s + (0.973 − 0.227i)37-s + ⋯
L(s)  = 1  + (0.582 − 0.812i)3-s + (−0.528 + 0.849i)5-s + (−0.321 − 0.946i)9-s + (0.935 − 0.352i)11-s + (−0.881 − 0.471i)13-s + (0.382 + 0.923i)15-s + (−0.608 − 0.793i)17-s + (0.729 + 0.683i)19-s + (0.896 + 0.442i)23-s + (−0.442 − 0.896i)25-s + (−0.956 − 0.290i)27-s + (0.634 + 0.773i)29-s + (0.258 + 0.965i)31-s + (0.258 − 0.965i)33-s + (0.973 − 0.227i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.780 - 0.624i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.780 - 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.278550788 - 0.7994668505i\)
\(L(\frac12)\) \(\approx\) \(2.278550788 - 0.7994668505i\)
\(L(1)\) \(\approx\) \(1.201080140 - 0.2279934826i\)
\(L(1)\) \(\approx\) \(1.201080140 - 0.2279934826i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.582 - 0.812i)T \)
5 \( 1 + (-0.528 + 0.849i)T \)
11 \( 1 + (0.935 - 0.352i)T \)
13 \( 1 + (-0.881 - 0.471i)T \)
17 \( 1 + (-0.608 - 0.793i)T \)
19 \( 1 + (0.729 + 0.683i)T \)
23 \( 1 + (0.896 + 0.442i)T \)
29 \( 1 + (0.634 + 0.773i)T \)
31 \( 1 + (0.258 + 0.965i)T \)
37 \( 1 + (0.973 - 0.227i)T \)
41 \( 1 + (0.555 + 0.831i)T \)
43 \( 1 + (-0.995 + 0.0980i)T \)
47 \( 1 + (-0.130 - 0.991i)T \)
53 \( 1 + (0.352 + 0.935i)T \)
59 \( 1 + (0.0327 - 0.999i)T \)
61 \( 1 + (-0.910 + 0.412i)T \)
67 \( 1 + (0.812 + 0.582i)T \)
71 \( 1 + (-0.980 + 0.195i)T \)
73 \( 1 + (0.751 + 0.659i)T \)
79 \( 1 + (-0.793 - 0.608i)T \)
83 \( 1 + (0.290 + 0.956i)T \)
89 \( 1 + (0.0654 - 0.997i)T \)
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

−19.989064986572820853449646242, −19.60332725906135301280509060355, −19.022284698274119669361468877606, −17.639057479474012688313896654778, −16.92129746544601987088613162889, −16.52917007151637620473361124528, −15.46176321360240742537282367609, −15.12976293628181390127062422713, −14.299001195820698488415660137212, −13.451368744789979762459800862538, −12.69011109242937967440057702227, −11.748076837738572805268942002195, −11.23312671230106963879113933312, −10.10587475346733013423747171017, −9.34433019558465434672031201775, −8.93762957151019565811107961655, −8.05629709296046461374197251505, −7.29551867744846147307773419112, −6.25071404441498185635904823517, −5.023360698556756288720355111333, −4.484956282389220679473340328880, −3.91586449073110210693409732486, −2.8164762593714332155806503069, −1.8738952103925641542783979757, −0.662517494640711785129665311691, 0.60373521551549644404950984215, 1.54444446466669839788532024524, 2.82919835611157903766987668707, 3.109433713811858189693975806948, 4.163498727563317384834002343911, 5.31812660751093096661718699752, 6.45120102238718719860111937375, 6.99168307266829543506536353070, 7.61300810222842129476864890117, 8.436067626869349940442782551257, 9.279255116320825445127703191527, 10.05571174827604355679716308129, 11.16202036168435559387931610705, 11.7787854109290445668120684973, 12.39142413995252507355475419143, 13.35537292235917483721357665528, 14.15541092094881648291803466781, 14.59692806935027331577160309802, 15.29364626269555270971904763205, 16.19660978478572336645083000894, 17.16166195784062161124517131459, 18.01429166980238026765331199656, 18.440798517389945299678383477984, 19.32473727712158902639375925661, 19.83958615345011663259494380669

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