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Properties

Label 1-1792-1792.139-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.999 + 0.0245i$
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  + (0.995 − 0.0980i)3-s + (−0.471 + 0.881i)5-s + (0.980 − 0.195i)9-s + (0.773 − 0.634i)11-s + (−0.881 + 0.471i)13-s + (−0.382 + 0.923i)15-s + (−0.382 − 0.923i)17-s + (0.956 + 0.290i)19-s + (0.831 + 0.555i)23-s + (−0.555 − 0.831i)25-s + (0.956 − 0.290i)27-s + (0.634 − 0.773i)29-s + (−0.707 − 0.707i)31-s + (0.707 − 0.707i)33-s + (−0.290 − 0.956i)37-s + ⋯
L(s)  = 1  + (0.995 − 0.0980i)3-s + (−0.471 + 0.881i)5-s + (0.980 − 0.195i)9-s + (0.773 − 0.634i)11-s + (−0.881 + 0.471i)13-s + (−0.382 + 0.923i)15-s + (−0.382 − 0.923i)17-s + (0.956 + 0.290i)19-s + (0.831 + 0.555i)23-s + (−0.555 − 0.831i)25-s + (0.956 − 0.290i)27-s + (0.634 − 0.773i)29-s + (−0.707 − 0.707i)31-s + (0.707 − 0.707i)33-s + (−0.290 − 0.956i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.999 + 0.0245i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.999 + 0.0245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.306044874 + 0.02830084896i\)
\(L(\frac12)\) \(\approx\) \(2.306044874 + 0.02830084896i\)
\(L(1)\) \(\approx\) \(1.475664990 + 0.05756733250i\)
\(L(1)\) \(\approx\) \(1.475664990 + 0.05756733250i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.995 - 0.0980i)T \)
5 \( 1 + (-0.471 + 0.881i)T \)
11 \( 1 + (0.773 - 0.634i)T \)
13 \( 1 + (-0.881 + 0.471i)T \)
17 \( 1 + (-0.382 - 0.923i)T \)
19 \( 1 + (0.956 + 0.290i)T \)
23 \( 1 + (0.831 + 0.555i)T \)
29 \( 1 + (0.634 - 0.773i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-0.290 - 0.956i)T \)
41 \( 1 + (0.555 - 0.831i)T \)
43 \( 1 + (0.995 + 0.0980i)T \)
47 \( 1 + (-0.923 + 0.382i)T \)
53 \( 1 + (0.634 + 0.773i)T \)
59 \( 1 + (0.881 + 0.471i)T \)
61 \( 1 + (0.0980 + 0.995i)T \)
67 \( 1 + (-0.0980 - 0.995i)T \)
71 \( 1 + (0.980 + 0.195i)T \)
73 \( 1 + (0.195 + 0.980i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (-0.290 + 0.956i)T \)
89 \( 1 + (0.831 - 0.555i)T \)
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

−20.02007832334422276101094602380, −19.74007917976700728286073639930, −19.052263709341467130029264534837, −17.9340854698267389534173587348, −17.26934153322278945866871091617, −16.38674421953790546588922422361, −15.73599479711841909796558429523, −14.855940394759681577224140225205, −14.54442532783583775539853999966, −13.44013787399334676833257479338, −12.696432817737368675140742219693, −12.29993095040363642413653943678, −11.25555562659652710075770650879, −10.18259535220379348831038616532, −9.49377184916730756312259852837, −8.79994753461732060856526244505, −8.18444221552237264418015443030, −7.32131536306985593425090591734, −6.68112855615483072054584542492, −5.16768800503920363958941841960, −4.66008969996152657005999887225, −3.77088305696090833267165432573, −2.97521617383557197795726017111, −1.86725406808590364215061591642, −1.02683569226863031844226585232, 0.89342652490626411044732111994, 2.22066979986196492890593284357, 2.85264688003117699912072398611, 3.70196954529681394227331450392, 4.358209096524364758857980099235, 5.5949219451014411468108127580, 6.737781238663513242362007486137, 7.29670176682321930283038846743, 7.851731106332077519670195040046, 9.03204673982411893084830231887, 9.41950380320201907612709270502, 10.34025089181354476190000508214, 11.37028927238369767887457988464, 11.84293054403803610860194950061, 12.8454057309631354862849059163, 13.95356706108726827233077748865, 14.124571036256512562451010712011, 14.91598582672798681265492741846, 15.67501271751085597674749862661, 16.3083157343349284445258784543, 17.38293963054944967731944405498, 18.23080226778558334901480510754, 18.9084889496062847666963337143, 19.51402514064471787793934003883, 19.91362702761537073369629444524

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