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Properties

Label 1-1792-1792.1749-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.999 + 0.0245i$
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  + (0.634 − 0.773i)3-s + (−0.956 + 0.290i)5-s + (−0.195 − 0.980i)9-s + (0.995 + 0.0980i)11-s + (−0.290 + 0.956i)13-s + (−0.382 + 0.923i)15-s + (0.382 + 0.923i)17-s + (−0.881 + 0.471i)19-s + (−0.555 + 0.831i)23-s + (0.831 − 0.555i)25-s + (−0.881 − 0.471i)27-s + (0.0980 + 0.995i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (0.471 − 0.881i)37-s + ⋯
L(s)  = 1  + (0.634 − 0.773i)3-s + (−0.956 + 0.290i)5-s + (−0.195 − 0.980i)9-s + (0.995 + 0.0980i)11-s + (−0.290 + 0.956i)13-s + (−0.382 + 0.923i)15-s + (0.382 + 0.923i)17-s + (−0.881 + 0.471i)19-s + (−0.555 + 0.831i)23-s + (0.831 − 0.555i)25-s + (−0.881 − 0.471i)27-s + (0.0980 + 0.995i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (0.471 − 0.881i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.999 + 0.0245i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.999 + 0.0245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001762014843 - 0.1435746787i\)
\(L(\frac12)\) \(\approx\) \(0.001762014843 - 0.1435746787i\)
\(L(1)\) \(\approx\) \(0.9741956738 - 0.1100844951i\)
\(L(1)\) \(\approx\) \(0.9741956738 - 0.1100844951i\)

Euler product

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

−20.282370555354998765363065048233, −19.80777290300677652650761262799, −19.149500262941832973243425257267, −18.37805103323144321411080824782, −17.05949745576246132114729627218, −16.754316122527149747597905938598, −15.7797247426630567666530982699, −15.233126144324678867725835572631, −14.69008656890618935720693763272, −13.809339617883467369905461862452, −13.01962422224153308576306567198, −11.99616860102798573989900626924, −11.494345791726342087795320556409, −10.50589197680745706172738577739, −9.82072853723242079117635467111, −8.96165746251339262192767471709, −8.28070039612732415538994087049, −7.701647265525320591968820079688, −6.69262441900687020814848102090, −5.5845449808995524815517316954, −4.4999081456060693881500199156, −4.206927342076302988988073970460, −3.142346931243774217086889048845, −2.484123033514110368835397400498, −0.99094495240340405141351364273, 0.025816832652295053881608754913, 1.367727184151568954742016310172, 1.99183193338001886180331078787, 3.294176112469410479664802665947, 3.78754548014105616916933860871, 4.65374362965080210029004478287, 6.17905265493237460285862888705, 6.65359990577301205602863497036, 7.48229059535795368457752716113, 8.17843278646666864268727372177, 8.867727825510178715074341540970, 9.66555106368333157885109495733, 10.76214274743975739906944249293, 11.634704319431105577241240067287, 12.242040249707741058598394458893, 12.75249585760196347117950581260, 13.93460647443816679844958329233, 14.490437403753970318709372103316, 14.94004888577529592088440799663, 15.91309605029251159119516307745, 16.75911360114523087620873460873, 17.52489000190314126209303852638, 18.37127357899904529098877228367, 19.24023322167902694245070979596, 19.45794791558987113890331734437

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