↑

Properties

Label 1-1792-1792.1563-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.877 - 0.480i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0327 + 0.999i)3-s + (−0.352 + 0.935i)5-s + (−0.997 − 0.0654i)9-s + (−0.227 + 0.973i)11-s + (0.773 + 0.634i)13-s + (−0.923 − 0.382i)15-s + (−0.130 + 0.991i)17-s + (0.812 − 0.582i)19-s + (0.659 + 0.751i)23-s + (−0.751 − 0.659i)25-s + (0.0980 − 0.995i)27-s + (0.290 − 0.956i)29-s + (0.965 − 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.412 − 0.910i)37-s + ⋯
L(s)  = 1  + (−0.0327 + 0.999i)3-s + (−0.352 + 0.935i)5-s + (−0.997 − 0.0654i)9-s + (−0.227 + 0.973i)11-s + (0.773 + 0.634i)13-s + (−0.923 − 0.382i)15-s + (−0.130 + 0.991i)17-s + (0.812 − 0.582i)19-s + (0.659 + 0.751i)23-s + (−0.751 − 0.659i)25-s + (0.0980 − 0.995i)27-s + (0.290 − 0.956i)29-s + (0.965 − 0.258i)31-s + (−0.965 − 0.258i)33-s + (−0.412 − 0.910i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.877 - 0.480i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.877 - 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5724035517 - 0.1465030837i\)
\(L(\frac12)\) \(\approx\) \(0.5724035517 - 0.1465030837i\)
\(L(1)\) \(\approx\) \(0.7632449782 + 0.4485703727i\)
\(L(1)\) \(\approx\) \(0.7632449782 + 0.4485703727i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.0327 + 0.999i)T \)
5 \( 1 + (-0.352 + 0.935i)T \)
11 \( 1 + (-0.227 + 0.973i)T \)
13 \( 1 + (0.773 + 0.634i)T \)
17 \( 1 + (-0.130 + 0.991i)T \)
19 \( 1 + (0.812 - 0.582i)T \)
23 \( 1 + (0.659 + 0.751i)T \)
29 \( 1 + (0.290 - 0.956i)T \)
31 \( 1 + (0.965 - 0.258i)T \)
37 \( 1 + (-0.412 - 0.910i)T \)
41 \( 1 + (-0.195 - 0.980i)T \)
43 \( 1 + (-0.881 - 0.471i)T \)
47 \( 1 + (-0.608 + 0.793i)T \)
53 \( 1 + (-0.973 - 0.227i)T \)
59 \( 1 + (-0.162 - 0.986i)T \)
61 \( 1 + (0.528 - 0.849i)T \)
67 \( 1 + (-0.999 - 0.0327i)T \)
71 \( 1 + (0.555 + 0.831i)T \)
73 \( 1 + (0.896 - 0.442i)T \)
79 \( 1 + (-0.991 + 0.130i)T \)
83 \( 1 + (-0.995 + 0.0980i)T \)
89 \( 1 + (-0.321 - 0.946i)T \)
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

−20.07978421817029963812636645499, −19.35847830232020929927693830135, −18.432103481048536632972713961661, −18.17742594088816498346691310313, −17.08325523492897679267383211305, −16.40633516210602814605777955737, −15.90157712217792111403188026569, −14.84214046884823540803117071439, −13.78909969017309097146814834054, −13.42066270903088857599970078411, −12.683985700786262372301251401074, −11.868110721212104111697831480197, −11.38392687832497769213594645662, −10.408660412646705425207486592981, −9.25393742192187459331042721004, −8.33822319022580841982141718382, −8.19852152433231493261960259325, −7.10225816519411693144674932494, −6.269312679176383960131788470486, −5.405133106802460282572999186233, −4.78943531951177418460284605651, −3.35538755416175875879650630266, −2.85194171245749852703842893251, −1.304425755448300308638789260609, −0.94956860436844443479072516957, 0.123672266365836927758310486754, 1.74558119027978760732562083756, 2.776797508479410164473602032962, 3.60848385861101994675919901298, 4.25523288975130253808389715172, 5.1403712302354983973184876973, 6.137459297057878008760797854482, 6.874706370892506602496049931370, 7.803790085177221083211051705, 8.6454757523000087477740168431, 9.61319756482054173215076207144, 10.10561590103722332666402222110, 11.05339199505095015633966492280, 11.39988518494639713903108163707, 12.333314986615570603938131200160, 13.48854679263107805954747655622, 14.176704523552078396007663519397, 14.942844313415716279464711833366, 15.687330031568282812393755693751, 15.81875808014020323339948823232, 17.25926693655488952321594156484, 17.49166721036842178135539052246, 18.55132768666332534666230409529, 19.30722760249270774890236887470, 19.95368334458317477419031170237

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