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Properties

Label 1-1792-1792.1027-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.780 + 0.624i$
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  + (−0.986 + 0.162i)3-s + (−0.227 − 0.973i)5-s + (0.946 − 0.321i)9-s + (−0.412 + 0.910i)11-s + (0.290 + 0.956i)13-s + (0.382 + 0.923i)15-s + (0.608 + 0.793i)17-s + (0.0327 + 0.999i)19-s + (0.442 − 0.896i)23-s + (−0.896 + 0.442i)25-s + (−0.881 + 0.471i)27-s + (−0.0980 + 0.995i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (−0.528 + 0.849i)37-s + ⋯
L(s)  = 1  + (−0.986 + 0.162i)3-s + (−0.227 − 0.973i)5-s + (0.946 − 0.321i)9-s + (−0.412 + 0.910i)11-s + (0.290 + 0.956i)13-s + (0.382 + 0.923i)15-s + (0.608 + 0.793i)17-s + (0.0327 + 0.999i)19-s + (0.442 − 0.896i)23-s + (−0.896 + 0.442i)25-s + (−0.881 + 0.471i)27-s + (−0.0980 + 0.995i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (−0.528 + 0.849i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.780 + 0.624i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.780 + 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1315917276 + 0.3750479891i\)
\(L(\frac12)\) \(\approx\) \(0.1315917276 + 0.3750479891i\)
\(L(1)\) \(\approx\) \(0.6570991176 + 0.06462906531i\)
\(L(1)\) \(\approx\) \(0.6570991176 + 0.06462906531i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.986 + 0.162i)T \)
5 \( 1 + (-0.227 - 0.973i)T \)
11 \( 1 + (-0.412 + 0.910i)T \)
13 \( 1 + (0.290 + 0.956i)T \)
17 \( 1 + (0.608 + 0.793i)T \)
19 \( 1 + (0.0327 + 0.999i)T \)
23 \( 1 + (0.442 - 0.896i)T \)
29 \( 1 + (-0.0980 + 0.995i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (-0.528 + 0.849i)T \)
41 \( 1 + (-0.831 + 0.555i)T \)
43 \( 1 + (0.634 - 0.773i)T \)
47 \( 1 + (-0.130 - 0.991i)T \)
53 \( 1 + (0.910 + 0.412i)T \)
59 \( 1 + (-0.683 - 0.729i)T \)
61 \( 1 + (-0.935 - 0.352i)T \)
67 \( 1 + (-0.162 - 0.986i)T \)
71 \( 1 + (-0.195 - 0.980i)T \)
73 \( 1 + (-0.659 + 0.751i)T \)
79 \( 1 + (-0.793 - 0.608i)T \)
83 \( 1 + (-0.471 + 0.881i)T \)
89 \( 1 + (-0.997 - 0.0654i)T \)
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

−19.621547288099197957024415412041, −19.09515909196555147214671272421, −18.315759083829225331326457462042, −17.81068081061858413931528864429, −17.14797988955510212826707137477, −15.99292908768456803444332251395, −15.76672453704213769122831383035, −14.864949482163113105515818231160, −13.7834986384622497026180688190, −13.308992301808754166167172388567, −12.31240891551371274746770775644, −11.49512306021595530497266876732, −10.98165751497050973701414311884, −10.40113194933851206104016840612, −9.54723929335338456699348949070, −8.37172571816158426970967052719, −7.391647837307579696433791794124, −7.039125932885754950340780362981, −5.813088953671412439378758551851, −5.57365480281519968955822059244, −4.42344497456588183186522242328, −3.29573927822047766311394790650, −2.7073882959613594515323180595, −1.25861918490157003884366467077, −0.18055538377515309130259311829, 1.27438670785513529631764624272, 1.894304911795933639147485113252, 3.59197697321369558376800717547, 4.339202005163782114338727549099, 4.997564598468086982014176677776, 5.75741009031586620652627912701, 6.6226532344247309621909239863, 7.482486091794530456622870884490, 8.36952551878815881548820743818, 9.24530231277413280137085810156, 10.05851155887437167812932594646, 10.69100076051046840617560338142, 11.70814170663539407608217835510, 12.301895519523500255616666595631, 12.74103305928337332625219653991, 13.64546559469528106763092102985, 14.79903680400575878217266514887, 15.426455790924044248624716117493, 16.34029942603872412373562538457, 16.80999097566456454009182631030, 17.23787325443287408509939505457, 18.495465336909177532304804114926, 18.65960676800009593277317940209, 19.89208438176873640120181581561, 20.67921274124898896147847320568

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