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Properties

Label 1-1792-1792.291-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.0878 + 0.996i$
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  + (0.412 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0878 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0878 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0878 + 0.996i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.0878 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178515703 + 1.079203630i\)
\(L(\frac12)\) \(\approx\) \(1.178515703 + 1.079203630i\)
\(L(1)\) \(\approx\) \(0.9415770104 + 0.2904742085i\)
\(L(1)\) \(\approx\) \(0.9415770104 + 0.2904742085i\)

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

−19.98060997360824668430053586954, −19.07085644440512726313673800363, −18.42639953274626015508445329528, −17.89117636406408085866157316985, −17.00544937555571748208309290074, −16.02140755713765640293867433826, −15.33904015240748494784260897946, −14.67635051348612417748351030049, −13.928996355657754098016721151200, −12.95653948744407564405362941798, −12.49837309404724746731484242039, −11.74080194959622483393717145485, −11.02956000326117387221079669017, −10.05281505498275706845927414159, −8.98026030588542052636101937785, −8.272191369882536847861607853983, −7.76154827598322698566285993792, −6.81124969517594702456762191384, −6.3524764642081784695356050873, −5.06527098638714481286042676323, −4.08386115361163605008571616903, −3.35645414559191680483893270733, −2.36991650709349053449545537922, −1.41902975653040340939217920765, −0.41687478117852694290960872874, 0.64957978198871066869993712948, 2.03677395636816645045507726461, 3.31370835850530318066117620255, 3.613726890634056544625055414455, 4.543590232941862384456297063659, 5.31057548109662410285700130687, 6.386171833665980129180892923555, 7.32369052560810495001942250221, 8.24980093900844791239169619328, 8.91274355183072289251956234466, 9.31437587203858306757721007404, 10.80134086772979745352584830962, 11.000900666385111804535731803290, 11.67077592703613754342099302025, 12.827570381619284511426470328856, 13.71962678171703239697993557416, 14.24947663088184078121674597322, 15.26649245027300827292255530429, 15.82259107032678035708414417280, 16.16695003521339752002353027689, 17.07969225534182888488601080853, 18.00780847795208165080286738380, 19.01844492606857769900568875592, 19.556137320014014482644890852661, 20.10295236664743644827052445470

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