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Properties

Label 1-1792-1792.459-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.999 + 0.0388i$
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  + (0.812 − 0.582i)3-s + (−0.849 + 0.528i)5-s + (0.321 − 0.946i)9-s + (0.352 − 0.935i)11-s + (−0.471 − 0.881i)13-s + (−0.382 + 0.923i)15-s + (0.608 − 0.793i)17-s + (−0.683 − 0.729i)19-s + (0.896 − 0.442i)23-s + (0.442 − 0.896i)25-s + (−0.290 − 0.956i)27-s + (−0.773 − 0.634i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.227 + 0.973i)37-s + ⋯
L(s)  = 1  + (0.812 − 0.582i)3-s + (−0.849 + 0.528i)5-s + (0.321 − 0.946i)9-s + (0.352 − 0.935i)11-s + (−0.471 − 0.881i)13-s + (−0.382 + 0.923i)15-s + (0.608 − 0.793i)17-s + (−0.683 − 0.729i)19-s + (0.896 − 0.442i)23-s + (0.442 − 0.896i)25-s + (−0.290 − 0.956i)27-s + (−0.773 − 0.634i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.227 + 0.973i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.999 + 0.0388i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.999 + 0.0388i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03290970400 - 1.694787033i\)
\(L(\frac12)\) \(\approx\) \(-0.03290970400 - 1.694787033i\)
\(L(1)\) \(\approx\) \(1.038346813 - 0.4998476212i\)
\(L(1)\) \(\approx\) \(1.038346813 - 0.4998476212i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.812 - 0.582i)T \)
5 \( 1 + (-0.849 + 0.528i)T \)
11 \( 1 + (0.352 - 0.935i)T \)
13 \( 1 + (-0.471 - 0.881i)T \)
17 \( 1 + (0.608 - 0.793i)T \)
19 \( 1 + (-0.683 - 0.729i)T \)
23 \( 1 + (0.896 - 0.442i)T \)
29 \( 1 + (-0.773 - 0.634i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (-0.227 + 0.973i)T \)
41 \( 1 + (0.555 - 0.831i)T \)
43 \( 1 + (-0.0980 + 0.995i)T \)
47 \( 1 + (-0.130 + 0.991i)T \)
53 \( 1 + (0.935 + 0.352i)T \)
59 \( 1 + (-0.999 + 0.0327i)T \)
61 \( 1 + (0.412 - 0.910i)T \)
67 \( 1 + (0.582 + 0.812i)T \)
71 \( 1 + (-0.980 - 0.195i)T \)
73 \( 1 + (0.751 - 0.659i)T \)
79 \( 1 + (0.793 - 0.608i)T \)
83 \( 1 + (-0.956 - 0.290i)T \)
89 \( 1 + (0.0654 + 0.997i)T \)
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

−20.2877718615920081561443927804, −19.64713225490532619958448655674, −19.25217016430579303676813662051, −18.42539027489571956270656669698, −17.09176703455359880083780110265, −16.709776957051939731668708437133, −15.934680713261883630778789088856, −14.93861375002066549832172569590, −14.832749082428470255617502294272, −13.88266124936906300691975778258, −12.7659213034326609365864645378, −12.37740784632280190245679640644, −11.40210770600262556147687039747, −10.51894590077622756763304713633, −9.73374135052754336583306789495, −8.95700357306980472894677820621, −8.421625373248507925185645498290, −7.457492821282469588226577593694, −6.959764192339330261249886160500, −5.48303443269528561419739805640, −4.66968635950170291476074906086, −3.9835741433786224825049067815, −3.403037123849013564915116270347, −2.11312038131077882824962718118, −1.37590536128456698810451672494, 0.30353987343216522631520635809, 0.9667721796506341786730567140, 2.494468031835322176990290197, 2.97972454292324053736347631228, 3.76798988702935443911022739796, 4.73826415617077496934389780757, 5.98106335709883151162706343579, 6.76803251006594511301097583766, 7.57592698792276560896929418079, 8.062113696345775335466812510153, 8.90518762999653564295205684757, 9.68844557513536498305553655943, 10.75328489590275589798818573790, 11.4540531649648548387445722856, 12.21346743325199783539386070559, 13.01814748771408211854414019679, 13.7123364225975947588222496446, 14.57623785032952151369207599736, 15.06574391739899809961971519044, 15.727441553808611425490264101583, 16.730629470914513009731573516940, 17.539917040587663622679127109216, 18.49690283058836790087730745339, 19.0414907427148053297982399423, 19.427024126438249744539543084682

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