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Properties

Label 1-1792-1792.1571-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.585 - 0.810i$
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  + (−0.412 − 0.910i)3-s + (0.999 + 0.0327i)5-s + (−0.659 + 0.751i)9-s + (0.162 − 0.986i)11-s + (−0.881 − 0.471i)13-s + (−0.382 − 0.923i)15-s + (0.991 − 0.130i)17-s + (−0.227 + 0.973i)19-s + (0.0654 + 0.997i)23-s + (0.997 + 0.0654i)25-s + (0.956 + 0.290i)27-s + (0.634 + 0.773i)29-s + (0.965 + 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.683 − 0.729i)37-s + ⋯
L(s)  = 1  + (−0.412 − 0.910i)3-s + (0.999 + 0.0327i)5-s + (−0.659 + 0.751i)9-s + (0.162 − 0.986i)11-s + (−0.881 − 0.471i)13-s + (−0.382 − 0.923i)15-s + (0.991 − 0.130i)17-s + (−0.227 + 0.973i)19-s + (0.0654 + 0.997i)23-s + (0.997 + 0.0654i)25-s + (0.956 + 0.290i)27-s + (0.634 + 0.773i)29-s + (0.965 + 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.683 − 0.729i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.585 - 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.498428328 - 0.7658605050i\)
\(L(\frac12)\) \(\approx\) \(1.498428328 - 0.7658605050i\)
\(L(1)\) \(\approx\) \(1.088688940 - 0.3314078506i\)
\(L(1)\) \(\approx\) \(1.088688940 - 0.3314078506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.412 - 0.910i)T \)
5 \( 1 + (0.999 + 0.0327i)T \)
11 \( 1 + (0.162 - 0.986i)T \)
13 \( 1 + (-0.881 - 0.471i)T \)
17 \( 1 + (0.991 - 0.130i)T \)
19 \( 1 + (-0.227 + 0.973i)T \)
23 \( 1 + (0.0654 + 0.997i)T \)
29 \( 1 + (0.634 + 0.773i)T \)
31 \( 1 + (0.965 + 0.258i)T \)
37 \( 1 + (-0.683 - 0.729i)T \)
41 \( 1 + (0.555 + 0.831i)T \)
43 \( 1 + (0.995 - 0.0980i)T \)
47 \( 1 + (0.793 - 0.608i)T \)
53 \( 1 + (-0.986 - 0.162i)T \)
59 \( 1 + (-0.849 - 0.528i)T \)
61 \( 1 + (0.812 + 0.582i)T \)
67 \( 1 + (0.910 - 0.412i)T \)
71 \( 1 + (0.980 - 0.195i)T \)
73 \( 1 + (-0.946 + 0.321i)T \)
79 \( 1 + (0.130 - 0.991i)T \)
83 \( 1 + (-0.290 - 0.956i)T \)
89 \( 1 + (-0.896 + 0.442i)T \)
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

−20.761355584485083473689934286652, −19.65337088204906408631789254670, −18.83203976928176059375168642578, −17.80175556835105811740505919560, −17.12251826830304430442206335589, −17.01892765493620694868670178998, −15.83495291981925735479737535765, −15.20790036190492703857719884406, −14.33914310984213855946414861913, −13.95100266481851839343731152130, −12.589394097878239118755318972135, −12.243789614120532716212680549148, −11.19768011977220687622966982891, −10.33644643642065781556198441067, −9.79638523583285526213107294254, −9.30249102023357634118724219, −8.36374826704493653749710829898, −7.11276611707753668398208842590, −6.414570558639358338755141294635, −5.57877571366239035836138869632, −4.71735837503671388551188729984, −4.29903729133565686521847969088, −2.88076366241265607327382031894, −2.2479691489834298068308548507, −0.90017755993444069188636866559, 0.84348740480017862699178351238, 1.6219106840350150185199608153, 2.62959622182937747041099581246, 3.38255887733910820948591243205, 4.92852222004151700145662569440, 5.64179568695677009916681189321, 6.10266108970936912860520419632, 7.0523024628795939103724869159, 7.831409376876228565647528300788, 8.60326965455704532319437363114, 9.61213929456385387267544174957, 10.36494208044736840295978662518, 11.08462495538800794079104045591, 12.11934465407328300985545408905, 12.5480642054766751127368477748, 13.425671468075108817981495952141, 14.16678838045261348486255431579, 14.47524586792350415826546537938, 15.88017457319625217897359011265, 16.658474349665807359503806523541, 17.28845613934091845652511986002, 17.77296317654183451223014014074, 18.68194674323520656501226277603, 19.1569970928809799036220929249, 19.92967720413517537418672113432

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