↑

Properties

Label 1-1792-1792.1179-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.996 + 0.0857i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.849 − 0.528i)3-s + (0.986 − 0.162i)5-s + (0.442 + 0.896i)9-s + (0.729 + 0.683i)11-s + (0.773 + 0.634i)13-s + (−0.923 − 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.910 − 0.412i)19-s + (0.321 − 0.946i)23-s + (0.946 − 0.321i)25-s + (0.0980 − 0.995i)27-s + (−0.290 + 0.956i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (0.582 − 0.812i)37-s + ⋯
L(s)  = 1  + (−0.849 − 0.528i)3-s + (0.986 − 0.162i)5-s + (0.442 + 0.896i)9-s + (0.729 + 0.683i)11-s + (0.773 + 0.634i)13-s + (−0.923 − 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.910 − 0.412i)19-s + (0.321 − 0.946i)23-s + (0.946 − 0.321i)25-s + (0.0980 − 0.995i)27-s + (−0.290 + 0.956i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (0.582 − 0.812i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.996 + 0.0857i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.996 + 0.0857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.657853416 + 0.07120232072i\)
\(L(\frac12)\) \(\approx\) \(1.657853416 + 0.07120232072i\)
\(L(1)\) \(\approx\) \(1.106173223 - 0.06306772969i\)
\(L(1)\) \(\approx\) \(1.106173223 - 0.06306772969i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.849 - 0.528i)T \)
5 \( 1 + (0.986 - 0.162i)T \)
11 \( 1 + (0.729 + 0.683i)T \)
13 \( 1 + (0.773 + 0.634i)T \)
17 \( 1 + (0.793 + 0.608i)T \)
19 \( 1 + (-0.910 - 0.412i)T \)
23 \( 1 + (0.321 - 0.946i)T \)
29 \( 1 + (-0.290 + 0.956i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (0.582 - 0.812i)T \)
41 \( 1 + (0.195 + 0.980i)T \)
43 \( 1 + (0.881 + 0.471i)T \)
47 \( 1 + (-0.991 - 0.130i)T \)
53 \( 1 + (-0.683 + 0.729i)T \)
59 \( 1 + (0.935 + 0.352i)T \)
61 \( 1 + (-0.999 - 0.0327i)T \)
67 \( 1 + (-0.528 + 0.849i)T \)
71 \( 1 + (0.555 + 0.831i)T \)
73 \( 1 + (0.0654 - 0.997i)T \)
79 \( 1 + (0.608 + 0.793i)T \)
83 \( 1 + (-0.995 + 0.0980i)T \)
89 \( 1 + (0.659 - 0.751i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.51874854783174743346760147044, −19.28752833463172944674782702512, −18.61759955173096374982970684335, −17.79760690343268375967708441771, −17.24768925874108181063298372227, −16.64026467231377580000442781506, −15.90293257287453035916396558848, −15.073640613444865213386680958202, −14.26032430251376573026595836629, −13.53191562285960504049047419796, −12.72299062913379727459300667866, −11.856305319662362506346763841051, −11.08395938514023810947279447789, −10.479586951366174099115906587080, −9.701227757391075552447887460967, −9.09599793357850252803277049714, −8.108986066008428011851797549799, −6.89462546460082445045130613917, −6.12931011910648264814399716732, −5.69273437178577320549052074725, −4.86379407787974294217996399693, −3.74573356527048797221673930233, −3.07719596195374596779891529579, −1.667919770249499065405516323, −0.814437852520265023306892925863, 1.062618374901229398954707284495, 1.70058689859145797737982107980, 2.58556195809721659316438202693, 4.09503606918218966543218264026, 4.75074756144028980836542853371, 5.80497714081874389241010662639, 6.33638084797155698992502656927, 6.91962792862439700599582518514, 7.99212322001383233401291950858, 8.95795044494556850855194722577, 9.67896269472231113303860099110, 10.59250189184551859972594013353, 11.14913294633236254284770313511, 12.12226232508973046780691810541, 12.82823552864789832864386662109, 13.25589299572113723408573487910, 14.33588348369082641216313988771, 14.80152010197837009664281305433, 16.13772490034669863154512818651, 16.74023093581313999908431670665, 17.227425079575549920875836951808, 17.97166544898165069630795956160, 18.59212866960707707153442286937, 19.309166378400245272504923747590, 20.22324026023762320627621823995

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