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Properties

Label 1-1792-1792.1115-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: $\chi_{1792} (1115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.877 + 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01991272725 - 0.07780120060i\)
\(L(\frac12)\) \(\approx\) \(0.01991272725 - 0.07780120060i\)
\(L(1)\) \(\approx\) \(1.057933060 - 0.1430394448i\)
\(L(1)\) \(\approx\) \(1.057933060 - 0.1430394448i\)

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

−20.39640478810273563228696047536, −19.75696575805097842155639670682, −18.81667709183937511833393127294, −18.634896677541202288943980734221, −17.77503040888380596044693639227, −16.35263722270349258597764298843, −15.992517246145740007179357316634, −15.23055135346018696355177279275, −14.60307299072393063115238029458, −13.786936202056031816371105227500, −13.17698726188357407051239365822, −12.24235609905747716092439451871, −11.51827062853539932223073163740, −10.62531445194062670792555685968, −9.86785321741691334725663184962, −8.979208679980172625564129127275, −8.21554357696670875024155346864, −7.58381134828506183272184116009, −6.986000532825669337599856988112, −5.91193447748186893156077823925, −4.62223523260989024345693689620, −4.02590944699608826567932231544, −3.16861864043782345884302903709, −2.408404873145895028198640675601, −1.38862906853401032237710094384, 0.0130549808733525957081787496, 1.03507450342725667228912455814, 2.288800422075127065279015497926, 3.03767261196477432709727606255, 3.9707107343388737120286658619, 4.507806107765417807420136329011, 5.64923814775041114981777458764, 6.67723210154316205400058681973, 7.74724399214524934701185404137, 8.17857832487725720016017849783, 8.67371888549114952145636647192, 9.71520492381988870851501726928, 10.61004237603061561181275146754, 11.1826883858050483002728466649, 12.37023715960017692293606630341, 13.06997803808982863110304800398, 13.379951376319523083356268406715, 14.67827339282754911122168596358, 15.112992356324131397122400655054, 15.831373452404032828319898564386, 16.32112043464400034466857826830, 17.4779799871108319772383017554, 18.32981855296257297706775743072, 19.04719699176679264738849370019, 19.58763132722036529286493441699

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