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Properties

Label 1-1792-1792.117-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.810 + 0.585i$
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

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.810 + 0.585i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.810 + 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1985777653 - 0.6137826458i\)
\(L(\frac12)\) \(\approx\) \(-0.1985777653 - 0.6137826458i\)
\(L(1)\) \(\approx\) \(1.089457252 - 0.4334307339i\)
\(L(1)\) \(\approx\) \(1.089457252 - 0.4334307339i\)

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

−20.56187791893143831267464341736, −19.92870126403930328322341411647, −19.14653479852903732812998391464, −17.940485403956319849649802450007, −17.65803729068611675590428387828, −16.63012857560866634192604307108, −16.18673428216659071613777404956, −15.027760123558944709088035473403, −14.658324401044041006255060734013, −14.020970121576193740195027255893, −13.0035161819993456186683162366, −12.450021071948106669897176578042, −11.29441397331671678096218120111, −10.37083601134387463651824696125, −9.93846585995065455754860472446, −9.33860695360394167206939638520, −8.51008552157189068056670191800, −7.51727751078774382336434202766, −6.7291934768111430836304291603, −5.48532875963704190239572821800, −5.06466461527591378542761248797, −4.251179838161321522784693627854, −2.9502597377235874652593664273, −2.54609448445443628601566756364, −1.42634593745451804534639763918, 0.094090062369537575209695111824, 1.332771563527345224782475371163, 1.892270037251471580772565084270, 2.946001497013294035801091462611, 3.5993642591006411576043944889, 5.13397187779183681745130359558, 5.771347730110090177625062799522, 6.47727045365919518310746905987, 7.38504295746910077555428583089, 8.10754585485335880744471907192, 8.92413974969814354320046545600, 9.71488105011604454850267514135, 10.3709221439531772145794832834, 11.595341332460105304357484367136, 12.14062212542318819970743392422, 13.03003281892322763609066312782, 13.65308139624154509462050387555, 14.29475019766884305283285964294, 14.737457877536094226469860301846, 16.03837272041542023853318983137, 16.83898370927320252859715860318, 17.41517894587904744797957491893, 18.17198069906155419695899575605, 18.99551777369111195230319276786, 19.26114269563513331425929396110

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