Properties

Label 1-1792-1792.69-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.219 + 0.975i$
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  + (−0.290 + 0.956i)3-s + (−0.0980 + 0.995i)5-s + (−0.831 − 0.555i)9-s + (0.881 + 0.471i)11-s + (−0.995 + 0.0980i)13-s + (−0.923 − 0.382i)15-s + (0.923 − 0.382i)17-s + (0.773 + 0.634i)19-s + (−0.195 − 0.980i)23-s + (−0.980 − 0.195i)25-s + (0.773 − 0.634i)27-s + (0.471 + 0.881i)29-s + (−0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (0.634 + 0.773i)37-s + ⋯
L(s)  = 1  + (−0.290 + 0.956i)3-s + (−0.0980 + 0.995i)5-s + (−0.831 − 0.555i)9-s + (0.881 + 0.471i)11-s + (−0.995 + 0.0980i)13-s + (−0.923 − 0.382i)15-s + (0.923 − 0.382i)17-s + (0.773 + 0.634i)19-s + (−0.195 − 0.980i)23-s + (−0.980 − 0.195i)25-s + (0.773 − 0.634i)27-s + (0.471 + 0.881i)29-s + (−0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (0.634 + 0.773i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.219 + 0.975i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.219 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246687581 + 1.557686849i\)
\(L(\frac12)\) \(\approx\) \(1.246687581 + 1.557686849i\)
\(L(1)\) \(\approx\) \(0.9018407127 + 0.4922545800i\)
\(L(1)\) \(\approx\) \(0.9018407127 + 0.4922545800i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.290 + 0.956i)T \)
5 \( 1 + (-0.0980 + 0.995i)T \)
11 \( 1 + (0.881 + 0.471i)T \)
13 \( 1 + (-0.995 + 0.0980i)T \)
17 \( 1 + (0.923 - 0.382i)T \)
19 \( 1 + (0.773 + 0.634i)T \)
23 \( 1 + (-0.195 - 0.980i)T \)
29 \( 1 + (0.471 + 0.881i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (0.634 + 0.773i)T \)
41 \( 1 + (0.980 - 0.195i)T \)
43 \( 1 + (-0.290 - 0.956i)T \)
47 \( 1 + (0.382 + 0.923i)T \)
53 \( 1 + (0.471 - 0.881i)T \)
59 \( 1 + (-0.995 - 0.0980i)T \)
61 \( 1 + (0.956 + 0.290i)T \)
67 \( 1 + (0.956 + 0.290i)T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (0.555 - 0.831i)T \)
79 \( 1 + (0.382 - 0.923i)T \)
83 \( 1 + (-0.634 + 0.773i)T \)
89 \( 1 + (0.195 - 0.980i)T \)
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

−19.671452804484585633528198369609, −19.33711703730836162967468324715, −18.26746425365862460108535512118, −17.50901557678331378031176869312, −16.95012808094165308933217171598, −16.40426804141065340273227349732, −15.46224281275703954368949389977, −14.34544393381328329078852734329, −13.8651509922701888034172414007, −12.94006652811455786427841316657, −12.365439732639206828611311030721, −11.748697497504807809875965972259, −11.12895591417378321136332792793, −9.79124515353484286983378174772, −9.21131775548450417775449478081, −8.23877986363869449146032539014, −7.641033597526953028819116636651, −6.86774170446415343337908543695, −5.77512779760223048871099395604, −5.35703786601516540541863262062, −4.30488851990692268417736206856, −3.25379530674572914331523462009, −2.14948508453763928701360671672, −1.16212047504665555117361337180, −0.6055829496579712210805923602, 0.69152563407894335901738314650, 2.1609985104243702674142854206, 3.0969167993263290859849052030, 3.80199682412026283551223363638, 4.65595849943607306826226173795, 5.531597840235711610143661883984, 6.388179255450631554902524642218, 7.17447758508609658760249308415, 7.98627648841522588836420978731, 9.199170400833566878544170945248, 9.79778664697920189302763592359, 10.33400728166323475626247227728, 11.19918707777473785045705462143, 11.93921565864109445645467779014, 12.43360938791625524707702019885, 13.98762186848886549217762850166, 14.52607936681083324284095364800, 14.84353276207117510116959172884, 15.82670618564974399488194783571, 16.57926200623769410953093948353, 17.145581114937706048833385788, 18.031857485229044480180870536058, 18.67412207593359125142108913029, 19.61153465626883858793970845992, 20.26858558301032707114060087418

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