Properties

Label 1-1792-1792.181-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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0980 − 0.995i)3-s + (0.881 + 0.471i)5-s + (−0.980 + 0.195i)9-s + (0.634 + 0.773i)11-s + (−0.471 − 0.881i)13-s + (0.382 − 0.923i)15-s + (−0.382 − 0.923i)17-s + (0.290 − 0.956i)19-s + (0.831 + 0.555i)23-s + (0.555 + 0.831i)25-s + (0.290 + 0.956i)27-s + (0.773 + 0.634i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (−0.956 + 0.290i)37-s + ⋯
L(s)  = 1  + (−0.0980 − 0.995i)3-s + (0.881 + 0.471i)5-s + (−0.980 + 0.195i)9-s + (0.634 + 0.773i)11-s + (−0.471 − 0.881i)13-s + (0.382 − 0.923i)15-s + (−0.382 − 0.923i)17-s + (0.290 − 0.956i)19-s + (0.831 + 0.555i)23-s + (0.555 + 0.831i)25-s + (0.290 + 0.956i)27-s + (0.773 + 0.634i)29-s + (0.707 + 0.707i)31-s + (0.707 − 0.707i)33-s + (−0.956 + 0.290i)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: $\chi_{1792} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.999 + 0.0245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.478157220 + 0.03041309126i\)
\(L(\frac12)\) \(\approx\) \(2.478157220 + 0.03041309126i\)
\(L(1)\) \(\approx\) \(1.216552160 - 0.2076135935i\)
\(L(1)\) \(\approx\) \(1.216552160 - 0.2076135935i\)

Euler product

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

−20.20691392510449846710219008728, −19.303920513559914422822854352352, −18.66194106730904298719724334961, −17.32778534808909803199058395836, −17.09542490037626601092987994084, −16.51643329383211395910108743328, −15.64758770195367706275176243906, −14.83545304878370357655140224736, −13.99373392680122276863789371495, −13.68430688218708934682687876862, −12.35246999521951626476331878671, −11.86501041804174631956313042183, −10.75135294858958587000800269523, −10.28387504555429768961774527445, −9.34308775120091219094675033931, −8.896961750297027873150677364597, −8.16026367025920527712577200968, −6.69580245450032354502607352148, −6.053682815851805824382006325845, −5.325651723379265627798090048206, −4.42179358873512628924861731905, −3.78209511113677076827959348696, −2.66302809785945346556260432666, −1.70116326195586262782952417523, −0.5374316503064430189245608154, 0.836846344935893999542559720706, 1.61474221504057430327641750903, 2.683241114475266227330206216350, 3.085459103299303577546845676056, 4.83331886992963324924461773011, 5.31743899540381500587601810655, 6.470504046648018142355715379896, 6.9243905385733603702593117651, 7.545332450052803498583209146805, 8.71774101512043225801717906758, 9.38684409368103671090762115059, 10.258956403797529958051752857282, 11.08174210537435662406275834143, 11.93534180784902775492735880002, 12.56834065187803360140153785869, 13.53991061574650003335986159064, 13.778023063463044707738753746763, 14.809950658120680265917231136486, 15.352200765061943739988682252255, 16.62856186536619482624808964371, 17.38230185319584004319694227419, 17.90291197179383667617541457081, 18.231132126154935934981618947131, 19.43508161414863579969024316607, 19.79462694323859437439364317123

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