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Properties

Label 1-1792-1792.507-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.765 - 0.643i$
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  + (0.227 + 0.973i)3-s + (0.582 − 0.812i)5-s + (−0.896 + 0.442i)9-s + (−0.999 + 0.0327i)11-s + (0.0980 + 0.995i)13-s + (0.923 + 0.382i)15-s + (−0.793 − 0.608i)17-s + (−0.352 − 0.935i)19-s + (0.946 + 0.321i)23-s + (−0.321 − 0.946i)25-s + (−0.634 − 0.773i)27-s + (0.881 − 0.471i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.162 − 0.986i)37-s + ⋯
L(s)  = 1  + (0.227 + 0.973i)3-s + (0.582 − 0.812i)5-s + (−0.896 + 0.442i)9-s + (−0.999 + 0.0327i)11-s + (0.0980 + 0.995i)13-s + (0.923 + 0.382i)15-s + (−0.793 − 0.608i)17-s + (−0.352 − 0.935i)19-s + (0.946 + 0.321i)23-s + (−0.321 − 0.946i)25-s + (−0.634 − 0.773i)27-s + (0.881 − 0.471i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.162 − 0.986i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.765 - 0.643i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.765 - 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.301215924 - 0.4746520195i\)
\(L(\frac12)\) \(\approx\) \(1.301215924 - 0.4746520195i\)
\(L(1)\) \(\approx\) \(1.085886500 + 0.06481631443i\)
\(L(1)\) \(\approx\) \(1.085886500 + 0.06481631443i\)

Euler product

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

−20.23086419025602322389027541276, −19.44376797296529767210315123592, −18.48412038004177036266226697256, −18.40858983711778627118214863125, −17.43389776018391896822551522074, −16.98132521761473937418694911592, −15.56682075875940144106603759739, −15.10461075162088779382441319359, −14.20701551028589200482618042339, −13.5867059432809486102070669695, −12.882340085464985310801446212421, −12.34991839467089703775297698398, −11.17924037477113188622704320119, −10.532577957976660722832545376, −9.938173671837121017596130884442, −8.50970372265834797862377760634, −8.294202773687272958018456579383, −7.11729224238828730223765620036, −6.65183533664406560033809544763, −5.768141738597152655704356970618, −5.068655079417400479199519224177, −3.52293657628873549951991289961, −2.82483252771089640670483922707, −2.12970651001860420231143729501, −1.10643532009747027729054867850, 0.49133839185629011032851437953, 2.08343543832841874538956123100, 2.64139923454364697851981764166, 3.87361146744423335327492049729, 4.8148683873464219501242101019, 5.05535578451593578238364957089, 6.14591873517430669305673086997, 7.10267989374302593783947022272, 8.3103082408757342122554127279, 8.84302535471875387584229277662, 9.53452300464930860704841797402, 10.16116159624048410581534681912, 11.13151574086062514326652178911, 11.67325034091897464692262485519, 12.925863009122638742475095177676, 13.47422031497278660091282279070, 14.09221096994872405273014840474, 15.15692604782961518723292702376, 15.738752105929543246232580799976, 16.34851102097019555426242846233, 17.1288949850545229962110569972, 17.682455333537019777209822749545, 18.69670304154215085159412748284, 19.59753966342684680708451356827, 20.27349542109715385047119145372

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