Properties

Label 1-1792-1792.1293-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.643 - 0.765i$
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  + (−0.973 − 0.227i)3-s + (0.812 − 0.582i)5-s + (0.896 + 0.442i)9-s + (0.0327 − 0.999i)11-s + (0.995 + 0.0980i)13-s + (−0.923 + 0.382i)15-s + (−0.793 + 0.608i)17-s + (0.935 + 0.352i)19-s + (0.946 − 0.321i)23-s + (0.321 − 0.946i)25-s + (−0.773 − 0.634i)27-s + (−0.471 + 0.881i)29-s + (−0.258 − 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.986 + 0.162i)37-s + ⋯
L(s)  = 1  + (−0.973 − 0.227i)3-s + (0.812 − 0.582i)5-s + (0.896 + 0.442i)9-s + (0.0327 − 0.999i)11-s + (0.995 + 0.0980i)13-s + (−0.923 + 0.382i)15-s + (−0.793 + 0.608i)17-s + (0.935 + 0.352i)19-s + (0.946 − 0.321i)23-s + (0.321 − 0.946i)25-s + (−0.773 − 0.634i)27-s + (−0.471 + 0.881i)29-s + (−0.258 − 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.986 + 0.162i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.643 - 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.049662851 - 0.9539996121i\)
\(L(\frac12)\) \(\approx\) \(2.049662851 - 0.9539996121i\)
\(L(1)\) \(\approx\) \(1.051223419 - 0.2356387358i\)
\(L(1)\) \(\approx\) \(1.051223419 - 0.2356387358i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.973 - 0.227i)T \)
5 \( 1 + (0.812 - 0.582i)T \)
11 \( 1 + (0.0327 - 0.999i)T \)
13 \( 1 + (0.995 + 0.0980i)T \)
17 \( 1 + (-0.793 + 0.608i)T \)
19 \( 1 + (0.935 + 0.352i)T \)
23 \( 1 + (0.946 - 0.321i)T \)
29 \( 1 + (-0.471 + 0.881i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.986 + 0.162i)T \)
41 \( 1 + (0.980 + 0.195i)T \)
43 \( 1 + (0.290 - 0.956i)T \)
47 \( 1 + (-0.991 + 0.130i)T \)
53 \( 1 + (0.999 + 0.0327i)T \)
59 \( 1 + (-0.412 + 0.910i)T \)
61 \( 1 + (0.729 + 0.683i)T \)
67 \( 1 + (0.227 - 0.973i)T \)
71 \( 1 + (0.831 + 0.555i)T \)
73 \( 1 + (-0.997 + 0.0654i)T \)
79 \( 1 + (0.608 - 0.793i)T \)
83 \( 1 + (0.634 + 0.773i)T \)
89 \( 1 + (0.751 - 0.659i)T \)
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

−20.34849101153507725157612705358, −19.25241566663011215963966470081, −18.23276477590952718298602332623, −17.93983739531747239544913983767, −17.4123199309200668502558843961, −16.457188384902980903092089973223, −15.73526798297190625036699484251, −15.09086321445079857842532207843, −14.20263916246154461904271483355, −13.22404496187471685729156708946, −12.852255066346546764787480190688, −11.598750647288318645604782278287, −11.1861057954252405261703941610, −10.43022394573304518971081492161, −9.548329756955842101889759173122, −9.19119652468523329353015184991, −7.6684101579116945147604602352, −6.891317396956169537471520563884, −6.33856169036329262563369864011, −5.43518065513176619630105904259, −4.80545190931765557160135726223, −3.79237299461416017964629073175, −2.72959667921030730319855662495, −1.6862266725350661263884961169, −0.764003839471029914693362319472, 0.690852356081347225877572407873, 1.24802360822138024004820336740, 2.27310405275331493954330219068, 3.5773057169411392204721584169, 4.52065707482381637824218512953, 5.46474284594338590861984073371, 5.9565654784421366027575414527, 6.60380772885648156613615389111, 7.66769314997025469113488317554, 8.670971717575539026481501584495, 9.27065058356943672122165891006, 10.29876345860254138165959689749, 11.00642472285508647934760252966, 11.53690764100780814978875078861, 12.59134831283339710196659397009, 13.23161457705850270787113972217, 13.631734021755058768736116209260, 14.71365917565230449389845016516, 15.79282717897788774334028898836, 16.49528777522556826089069453255, 16.814423216324848955466333412306, 17.780496929390273314045923221727, 18.297767021463930371356451449261, 18.94494981254195731565945120058, 19.975511043430828374833627443185

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